The Central Idea
A limit at infinity asks what happens to \(f(x)\) as \(x\) becomes arbitrarily large in magnitude.
To study the right end of a graph, we write
\[ \boxed{\lim_{x\to+\infty}f(x)}. \]To study the left end, we write
\[ \boxed{\lim_{x\to-\infty}f(x)}. \]These are statements about long-run behavior rather than behavior near one finite input.
Infinity Describes Direction, Not an Input Value
In
\[ x\to+\infty, \]the variable \(x\) does not eventually arrive at a real number called infinity.
Instead, \(x\) is allowed to become larger than any chosen positive bound.
Likewise,
\[ x\to-\infty \]means that \(x\) becomes more and more negative without bound.
Limits at Infinity Are About End Behavior
The phrase end behavior describes what a graph does far to the right and far to the left.
A function may:
- approach a finite number;
- grow toward \(+\infty\);
- decrease toward \(-\infty\);
- oscillate without settling;
- behave differently in the two directions.
A First Example
Consider
\[ f(x)=\frac{1}{x}. \]As \(x\) becomes very large and positive,
\[ \frac{1}{x} \]becomes very small and positive.
Therefore,
\[ \boxed{\lim_{x\to+\infty}\frac{1}{x}=0}. \]The Same Function to the Left
As \(x\) becomes very large in magnitude and negative,
\[ \frac{1}{x} \]becomes very small and negative.
It still approaches \(0\).
\[ \boxed{\lim_{x\to-\infty}\frac{1}{x}=0}. \]Numerical Evidence
For
\[ f(x)=\frac{1}{x}, \]a table illustrates the trend:
| \(x\) | \(\frac{1}{x}\) |
|---|---|
| \(10\) | \(0.1\) |
| \(100\) | \(0.01\) |
| \(1000\) | \(0.001\) |
| \(-10\) | \(-0.1\) |
| \(-100\) | \(-0.01\) |
| \(-1000\) | \(-0.001\) |
The outputs approach \(0\) in both directions.
Horizontal Asymptotes
If
\[ \lim_{x\to+\infty}f(x)=L \]or
\[ \lim_{x\to-\infty}f(x)=L, \]then the horizontal line
\[ \boxed{y=L} \]is a horizontal asymptote of the function in that direction.
A Function Can Cross a Horizontal Asymptote
A horizontal asymptote describes end behavior.
It does not create a barrier that the graph is forbidden to cross.
A function may cross the line \(y=L\) many times and still satisfy
\[ \lim_{x\to+\infty}f(x)=L. \]Do Not Confuse Horizontal and Vertical Asymptotes
A vertical asymptote describes behavior near a finite input:
\[ x=a. \]A horizontal asymptote describes behavior as the input moves without bound:
\[ y=L. \]These are different geometric features produced by different kinds of limits.
Reciprocal Powers
For any positive integer \(n\),
\[ \boxed{\lim_{x\to+\infty}\frac{1}{x^n}=0}. \]The same is true as \(x\to-\infty\):
\[ \boxed{\lim_{x\to-\infty}\frac{1}{x^n}=0}. \]The sign may differ while approaching \(0\), but the limit itself is still \(0\).
Why Reciprocal Powers Go to Zero
As \(|x|\) becomes very large,
\[ |x|^n \]becomes very large.
Therefore,
\[ \frac{1}{|x|^n} \]becomes arbitrarily small.
This principle is one of the main tools for rational limits at infinity.
Polynomial End Behavior
For a polynomial, the highest-degree term eventually dominates the lower degree terms.
Consider
\[ f(x)=3x^4-7x^2+2x-10. \]For very large \(|x|\), the term
\[ 3x^4 \]has the greatest effect on the overall value.
Dominant-Term Reasoning
If
\[ f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0, \]then for large \(|x|\),
\[ f(x) \]behaves approximately like
\[ a_nx^n. \]This is why polynomial end behavior can usually be predicted from the leading term alone.
Even-Degree Polynomials
Suppose the leading term is
\[ 2x^4. \]Since \(x^4\) is positive for both large positive and large negative \(x\),
\[ \boxed{\lim_{x\to+\infty}2x^4=+\infty} \]and
\[ \boxed{\lim_{x\to-\infty}2x^4=+\infty}. \]Negative Even-Degree Leading Terms
For
\[ -5x^6, \]the even power is positive but the coefficient is negative.
Therefore,
\[ \boxed{\lim_{x\to+\infty}-5x^6=-\infty} \]and
\[ \boxed{\lim_{x\to-\infty}-5x^6=-\infty}. \]Odd-Degree Polynomials
For the leading term
\[ 3x^5, \]the sign changes between the two ends.
\[ \boxed{\lim_{x\to+\infty}3x^5=+\infty} \]while
\[ \boxed{\lim_{x\to-\infty}3x^5=-\infty}. \]Negative Odd-Degree Leading Terms
For
\[ -2x^3, \]the directions reverse:
\[ \boxed{\lim_{x\to+\infty}-2x^3=-\infty} \]and
\[ \boxed{\lim_{x\to-\infty}-2x^3=+\infty}. \]Rational Functions
A rational function has the form
\[ R(x)=\frac{P(x)}{Q(x)}, \]where \(P(x)\) and \(Q(x)\) are polynomials.
At infinity, the most important comparison is usually between the degrees of the numerator and denominator.
Case 1: Numerator Degree Less Than Denominator Degree
Consider
\[ \frac{3x+1}{x^2+5}. \]The numerator has degree \(1\), while the denominator has degree \(2\).
The denominator grows faster.
Therefore,
\[ \boxed{\lim_{x\to\pm\infty}\frac{3x+1}{x^2+5}=0}. \]Why the Larger Denominator Degree Wins
Divide the numerator and denominator by \(x^2\):
\[ \frac{3x+1}{x^2+5} = \frac{\frac{3}{x}+\frac{1}{x^2}} {1+\frac{5}{x^2}}. \]As \(x\to\pm\infty\),
\[ \frac{1}{x}\to0 \qquad\text{and}\qquad \frac{1}{x^2}\to0. \]Thus,
\[ \frac{0+0}{1+0}=0. \]Case 2: Equal Degrees
Consider
\[ \frac{4x^2-3x+1}{2x^2+5x-7}. \]Both numerator and denominator have degree \(2\).
For large \(|x|\), the function behaves like
\[ \frac{4x^2}{2x^2}=2. \]Therefore,
\[ \boxed{ \lim_{x\to\pm\infty} \frac{4x^2-3x+1}{2x^2+5x-7} =2 }. \]Equal Degrees: Leading-Coefficient Rule
If the numerator and denominator have the same degree \(n\),
\[ \frac{a_nx^n+\cdots}{b_nx^n+\cdots}, \]then
\[ \boxed{ \lim_{x\to\pm\infty} \frac{a_nx^n+\cdots}{b_nx^n+\cdots} = \frac{a_n}{b_n} }. \]The horizontal asymptote is
\[ \boxed{y=\frac{a_n}{b_n}}. \]Case 3: Numerator Degree Greater Than Denominator Degree
Consider
\[ \frac{x^3+1}{x^2+4}. \]The numerator grows faster than the denominator.
For large \(|x|\), the function behaves roughly like
\[ \frac{x^3}{x^2}=x. \]Therefore,
\[ \boxed{\lim_{x\to+\infty}\frac{x^3+1}{x^2+4}=+\infty} \]and
\[ \boxed{\lim_{x\to-\infty}\frac{x^3+1}{x^2+4}=-\infty}. \]No Finite Horizontal Asymptote in the Higher-Numerator Case
If the numerator degree exceeds the denominator degree, the rational function does not approach a finite constant simply because of degree balance.
It may instead grow without bound or approach a nonhorizontal asymptotic shape.
Polynomial division can reveal that larger-scale structure.
The Degree Comparison Shortcut
For
\[ \frac{P(x)}{Q(x)}, \]compare
\[ \deg P \qquad\text{and}\qquad \deg Q. \]- If \(\deg P<\deg Q\), the limit is \(0\).
- If \(\deg P=\deg Q\), use the ratio of leading coefficients.
- If \(\deg P>\deg Q\), inspect the dominant quotient or divide the polynomials.
Why Dividing by the Highest Power Works
Consider
\[ \frac{5x^3-2x}{2x^3+x^2+1}. \]Divide every term by \(x^3\):
\[ \frac{5-\frac{2}{x^2}} {2+\frac{1}{x}+\frac{1}{x^3}}. \]As \(x\to\pm\infty\), every reciprocal power tends to \(0\).
Therefore,
\[ \boxed{ \lim_{x\to\pm\infty} \frac{5x^3-2x}{2x^3+x^2+1} = \frac{5}{2} }. \]Example: Lower Numerator Degree
Evaluate
\[ \lim_{x\to+\infty}\frac{7x^2+3}{2x^5-x}. \]Divide by \(x^5\):
\[ \frac{\frac{7}{x^3}+\frac{3}{x^5}} {2-\frac{1}{x^4}}. \]Every reciprocal power approaches \(0\), so
\[ \boxed{0}. \]Example: Equal Degrees with Negative Coefficients
Evaluate
\[ \lim_{x\to-\infty} \frac{-6x^4+2x}{3x^4-x^2+8}. \]The degrees are equal, so use the leading coefficients:
\[ \boxed{\frac{-6}{3}=-2}. \]Example: One Degree Higher in the Numerator
Consider
\[ \frac{2x^2+1}{x-3}. \]The dominant quotient is
\[ \frac{2x^2}{x}=2x. \]Thus,
\[ \lim_{x\to+\infty}\frac{2x^2+1}{x-3}=+\infty \]and
\[ \lim_{x\to-\infty}\frac{2x^2+1}{x-3}=-\infty. \]Polynomial Division Can Reveal Slant Behavior
For the function
\[ \frac{x^2+1}{x}, \]division gives
\[ x+\frac{1}{x}. \]Since
\[ \frac{1}{x}\to0, \]the graph becomes increasingly close to the line
\[ y=x. \]This is an example of nonhorizontal asymptotic behavior.
Radical Functions at Infinity
Radicals require attention to both dominant terms and signs.
Consider
\[ \sqrt{x^2+1}. \]Factor \(x^2\) inside the radical:
\[ \sqrt{x^2\left(1+\frac{1}{x^2}\right)} = |x|\sqrt{1+\frac{1}{x^2}}. \]The absolute value is essential because
\[ \sqrt{x^2}=|x|. \]Right-End Radical Behavior
As \(x\to+\infty\),
\[ |x|=x. \]Therefore,
\[ \sqrt{x^2+1} = x\sqrt{1+\frac{1}{x^2}}. \]Since the square-root factor approaches \(1\),
\[ \boxed{\lim_{x\to+\infty}\sqrt{x^2+1}=+\infty}. \]Left-End Radical Behavior
As \(x\to-\infty\),
\[ |x|=-x. \]Thus,
\[ \sqrt{x^2+1} = -x\sqrt{1+\frac{1}{x^2}}. \]Because \(-x\to+\infty\),
\[ \boxed{\lim_{x\to-\infty}\sqrt{x^2+1}=+\infty}. \]A Useful Radical Difference
Consider
\[ \sqrt{x^2+1}-x \]as \(x\to+\infty\).
Direct comparison gives an expression of the form
\[ \infty-\infty, \]which does not determine the answer.
Rationalize:
\[ \sqrt{x^2+1}-x = \frac{1} {\sqrt{x^2+1}+x}. \]The denominator grows without bound, so
\[ \boxed{ \lim_{x\to+\infty} \left(\sqrt{x^2+1}-x\right) =0 }. \]Exponential Growth at Infinity
For a base
\[ b>1, \]exponential functions grow without bound:
\[ \boxed{\lim_{x\to+\infty}b^x=+\infty}. \]In the opposite direction,
\[ \boxed{\lim_{x\to-\infty}b^x=0}. \]Exponential Decay
If
\[ 0and
\[ \boxed{\lim_{x\to-\infty}b^x=+\infty}. \]Logarithmic Growth
For the natural logarithm,
\[ \boxed{\lim_{x\to+\infty}\ln x=+\infty}. \]The growth is unbounded but much slower than polynomial and exponential growth.
The real logarithm is defined only for
\[ x>0, \]so there is no corresponding real limit as \(x\to-\infty\).
Growth Rates Matter
Many limits at infinity are governed by which part of an expression grows faster.
A useful long-run hierarchy is:
\[ \text{logarithmic} \;<\; \text{power} \;<\; \text{exponential} \]in the sense that, under suitable positive-domain conditions, exponentials eventually outgrow powers, and powers eventually outgrow logarithms.
Example: Exponential Dominates Polynomial
Consider
\[ \frac{x^3}{2^x}. \]The exponential denominator eventually grows much faster than the polynomial numerator.
Therefore,
\[ \boxed{\lim_{x\to+\infty}\frac{x^3}{2^x}=0}. \]Later analysis methods will provide systematic proofs of limits like this.
Graphical Interpretation
To estimate a limit at infinity from a graph, follow the curve farther and farther to the right or left.
Ask whether the graph:
- levels off toward a horizontal line;
- rises without bound;
- falls without bound;
- approaches a slanted or curved path;
- continues oscillating.
Numerical Tables for End Behavior
A useful numerical table should use increasingly large magnitudes.
For \(x\to+\infty\), sample values such as
\[ 10,\quad100,\quad1000,\quad10{,}000. \]For \(x\to-\infty\), use
\[ -10,\quad-100,\quad-1000,\quad-10{,}000. \]Random nearby values are less useful for long-run analysis.
A Function Can Have Different End Limits
Consider
\[ f(x)=\frac{x}{\sqrt{x^2+1}}. \]For large positive \(x\),
\[ f(x)\to1. \]For large negative \(x\),
\[ f(x)\to-1. \]Thus,
\[ \boxed{\lim_{x\to+\infty}\frac{x}{\sqrt{x^2+1}}=1} \]and
\[ \boxed{\lim_{x\to-\infty}\frac{x}{\sqrt{x^2+1}}=-1}. \]Why the Absolute Value Matters Again
Rewrite
\[ \sqrt{x^2+1} = |x|\sqrt{1+\frac{1}{x^2}}. \]Then
\[ \frac{x}{\sqrt{x^2+1}} = \frac{x}{|x|} \cdot \frac{1}{\sqrt{1+\frac{1}{x^2}}}. \]The factor
\[ \frac{x}{|x|} \]equals \(1\) for positive \(x\) and \(-1\) for negative \(x\).
That sign difference controls the two end limits.
Some Functions Have No Limit at Infinity
Consider
\[ f(x)=\sin x. \]As \(x\to+\infty\), the function continues oscillating between
\[ -1 \qquad\text{and}\qquad 1. \]It does not settle toward one finite value or grow consistently toward one infinite direction.
Therefore,
\[ \boxed{\lim_{x\to+\infty}\sin x\text{ does not exist}}. \]Bounded Does Not Automatically Mean Convergent
The function
\[ \sin x \]remains bounded:
\[ -1\le\sin x\le1. \]But boundedness alone does not force a limit to exist.
A limit requires the function values to approach one common target.
A Reliable Procedure for Rational Limits at Infinity
- Identify the highest degree in the numerator.
- Identify the highest degree in the denominator.
- Compare the degrees.
- If helpful, divide every term by the highest relevant power of \(x\).
- Use \(\frac{1}{x^n}\to0\) for positive integers \(n\).
- Track signs separately for \(x\to+\infty\) and \(x\to-\infty\).
- Interpret any finite constant as a horizontal asymptote.
A Reliable Procedure for General Limits at Infinity
- Identify the direction: \(+\infty\) or \(-\infty\).
- Look for the dominant part of the expression.
- Factor or divide by the dominant power when appropriate.
- Use reciprocal-power limits to eliminate vanishing terms.
- Check for absolute values when square roots of even powers appear.
- Use known growth-rate comparisons when exponentials or logarithms appear.
- Confirm the result graphically or numerically when useful.
Example: Full Rational Workflow
Evaluate
\[ \lim_{x\to+\infty} \frac{6x^3-4x+1}{3x^3+2x^2-5}. \]The numerator and denominator both have degree \(3\).
Divide by \(x^3\):
\[ \frac{6-\frac{4}{x^2}+\frac{1}{x^3}} {3+\frac{2}{x}-\frac{5}{x^3}}. \]Every reciprocal-power term approaches \(0\).
Therefore,
\[ \boxed{2}. \]The graph has the horizontal asymptote
\[ \boxed{y=2}. \]Example: Different Degrees
Evaluate
\[ \lim_{x\to-\infty} \frac{5x^2+1}{x^4+3}. \]The denominator degree is larger.
Divide by \(x^4\):
\[ \frac{\frac{5}{x^2}+\frac{1}{x^4}} {1+\frac{3}{x^4}}. \]Every reciprocal-power term tends to \(0\), so
\[ \boxed{0}. \]Example: Polynomial End Behavior
Evaluate
\[ \lim_{x\to-\infty} \left(-4x^5+2x^2-7\right). \]The dominant term is
\[ -4x^5. \]As \(x\to-\infty\),
\[ x^5\to-\infty, \]so multiplying by \(-4\) gives
\[ \boxed{+\infty}. \]Example: Radical Ratio
Evaluate
\[ \lim_{x\to+\infty} \frac{3x}{\sqrt{9x^2+4}}. \]Factor \(x^2\) inside the radical:
\[ \sqrt{9x^2+4} = x\sqrt{9+\frac{4}{x^2}} \]because \(x>0\) for sufficiently large positive \(x\).
Therefore,
\[ \frac{3x}{\sqrt{9x^2+4}} = \frac{3}{\sqrt{9+\frac{4}{x^2}}} \to \frac{3}{3} = \boxed{1}. \]Common Error: Substituting Infinity as a Number
Expressions such as
\[ \frac{1}{\infty} \]are informal shorthand, not ordinary arithmetic.
The precise statement is
\[ \lim_{x\to+\infty}\frac{1}{x}=0. \]Use limit reasoning rather than treating infinity as a real input.
Common Error: Comparing Coefficients Before Degrees
In
\[ \frac{1000x}{x^2}, \]the large coefficient \(1000\) does not control the end behavior.
The denominator has the larger degree:
\[ \frac{1000x}{x^2}=\frac{1000}{x}\to0. \]Degree growth eventually dominates fixed coefficients.
Common Error: Using the Leading-Coefficient Ratio for Unequal Degrees
The ratio of leading coefficients applies directly only when the numerator and denominator have the same degree.
For
\[ \frac{3x^2}{5x^4}, \]the answer is not
\[ \frac{3}{5}. \]Instead,
\[ \frac{3x^2}{5x^4} = \frac{3}{5x^2} \to0. \]Common Error: Forgetting the Left-End Sign
Odd powers change sign when \(x\) becomes negative.
For example,
\[ x^3\to-\infty \]as
\[ x\to-\infty. \]The left end of a graph must therefore be analyzed separately from the right when odd powers are involved.
Common Error: Replacing \(\sqrt{x^2}\) with \(x\)
The correct identity is
\[ \boxed{\sqrt{x^2}=|x|}. \]When \(x\to-\infty\),
\[ |x|=-x. \]Forgetting the absolute value can reverse the sign of a left-end limit.
Common Error: Assuming Bounded Oscillation Has a Limit
A function can remain within a fixed range and still fail to approach one value.
The standard example is
\[ \sin x. \]Boundedness and convergence are different properties.
Practice 1: Reciprocal Power
Evaluate:
\[ \lim_{x\to+\infty}\frac{1}{x^4}. \]Practice 2: Polynomial End Behavior
Evaluate:
\[ \lim_{x\to-\infty}(3x^6-2x+1). \]Practice 3: Odd-Degree Polynomial
Evaluate:
\[ \lim_{x\to-\infty}(5x^3-x). \]Practice 4: Lower Numerator Degree
Evaluate:
\[ \lim_{x\to+\infty} \frac{4x+7}{x^2+1}. \]Practice 5: Equal Degrees
Evaluate:
\[ \lim_{x\to-\infty} \frac{8x^3-x}{2x^3+5}. \]Practice 6: Higher Numerator Degree
Evaluate:
\[ \lim_{x\to+\infty} \frac{x^4+1}{x^2+3}. \]Practice 7: Horizontal Asymptote
Find the horizontal asymptote of
\[ f(x)=\frac{5x^2+4}{2x^2-7x+1}. \]Practice 8: Radical Ratio
Evaluate:
\[ \lim_{x\to+\infty} \frac{x}{\sqrt{x^2+9}}. \]Practice 9: Left-End Radical Ratio
Evaluate:
\[ \lim_{x\to-\infty} \frac{x}{\sqrt{x^2+9}}. \]Practice 10: Oscillation
Evaluate:
\[ \lim_{x\to+\infty}\cos x. \]Practice Answers
Practice 1
\[ \boxed{\lim_{x\to+\infty}\frac{1}{x^4}=0}. \]Practice 2
The dominant term is \(3x^6\), and an even power is positive on both ends:
\[ \boxed{+\infty}. \]Practice 3
The dominant term is \(5x^3\). As \(x\to-\infty\),
\[ 5x^3\to-\infty. \]Therefore,
\[ \boxed{-\infty}. \]Practice 4
The denominator degree is larger, so
\[ \boxed{0}. \]Practice 5
The degrees are equal. Use the ratio of leading coefficients:
\[ \boxed{\frac{8}{2}=4}. \]Practice 6
The function behaves like
\[ \frac{x^4}{x^2}=x^2. \]Therefore,
\[ \boxed{+\infty}. \]Practice 7
The degrees are equal, so the horizontal asymptote is the ratio of leading coefficients:
\[ \boxed{y=\frac{5}{2}}. \]Practice 8
For large positive \(x\),
\[ \sqrt{x^2+9} = x\sqrt{1+\frac{9}{x^2}}. \]Thus,
\[ \frac{x}{\sqrt{x^2+9}} = \frac{1}{\sqrt{1+\frac{9}{x^2}}} \to \boxed{1}. \]Practice 9
For large negative \(x\),
\[ \sqrt{x^2+9} = -x\sqrt{1+\frac{9}{x^2}}. \]Therefore,
\[ \frac{x}{\sqrt{x^2+9}} = -\frac{1}{\sqrt{1+\frac{9}{x^2}}} \to \boxed{-1}. \]Practice 10
The cosine function continues oscillating between \(-1\) and \(1\).
Therefore,
\[ \boxed{\lim_{x\to+\infty}\cos x\text{ does not exist}}. \]What to Remember
Limits at infinity describe long-run behavior:
\[ \boxed{\lim_{x\to+\infty}f(x)} \qquad\text{and}\qquad \boxed{\lim_{x\to-\infty}f(x)}. \]- Reciprocal powers \(\frac{1}{x^n}\) approach \(0\).
- Polynomial end behavior is governed by the leading term.
- For rational functions, compare numerator and denominator degrees.
- Equal rational degrees produce the ratio of leading coefficients.
- A finite limit at infinity produces a horizontal asymptote.
- Square roots of \(x^2\) require \(|x|\), especially as \(x\to-\infty\).
- Bounded oscillation does not guarantee a limit.
- Different ends of the same function may have different limits.
The central strategic question is: which part of the expression controls the function when \(|x|\) becomes very large?
Where We Go Next
Limits at infinity often become manageable by identifying dominant terms. Other limit problems produce expressions such as \(0/0\), \(\infty/\infty\), or \(\infty-\infty\), where the visible form does not determine the answer.
The next lesson, Indeterminate Forms, explains why these forms require further analysis and develops the algebraic viewpoint needed to resolve them.