Infinite Limits & Limits at Infinity (College Board AP® Calculus AB): Revision Note

Infinite limits

What is an infinite limit?

• Sometimes the values of a function become unbounded (in the positive or negative direction) as x approaches a certain value

  • In such cases we talk about the function having an infinite limit at that value of x

  • For functions defined as fractions or quotients, this happens when the denominator becomes 0 for some value(s) of x

• If the value of a function f increases without bound as x approaches some value c, then we write $\underset{x\to c}{\lim }f\left(x\right)=\infty$.

  • For example, $\underset{x\to 0}{\lim }\frac{1}{x^2}=\infty$

• If the value of a function f decreases without bound as x approaches some value c, then we write $\underset{x\to c}{\lim }f\left(x\right)=-\infty$.

  • For example, $\underset{x\to 0}{\lim }\left(-\frac{1}{x^2}\right)=-\infty$

• It is possible for the one-sided limits to be different

  • For example, $\underset{x\to 0^{+}}{\lim }\frac{1}{x}=\infty$

  • But $\underset{x\to 0^{-}}{\lim }\frac{1}{x}=-\infty$

How about limits that look like 0/∞ or ∞/0?

• Sometimes substitution can give limits that look like $\frac{0}{\pm \infty }$ or $\frac{\pm \infty }{0}$

• In the first case the limit will just be equal to zero

  • For example $\underset{x\to 0^{+}}{\lim }\frac{x}{\ln x}$

  • As $x$ approaches zero through the positive numbers

    • $x\to 0$

    • and $\ln x\to -\infty$

  • $x$ is a positive number close to zero and $\ln x$ is a large negative number

  • So near $x=0$ 'from the right', $\frac{x}{\ln x}$ is negative and getting closer and closer to zero

    • Therefore $\underset{x\to 0^{+}}{\lim }\frac{x}{\ln x}=0$

• In the second case the limit will diverge to either $+\infty$ or $-\infty$ depending on the behavior near the limit point

  • For example $\underset{x\to 0^{+}}{\lim }\frac{\ln x}{x}$

  • As $x$ approaches zero through the positive numbers

    • $\ln x\to -\infty$

    • and $x\to 0$

  • $\ln x$ is a large negative number and $x$ is a positive number close to zero

  • So near $x=0$ 'from the right', $\frac{\ln x}{x}$ is negative and getting more and more negative (i.e. bigger and bigger in the negative direction)

    • Therefore $\underset{x\to 0^{+}}{\lim }\frac{\ln x}{x}=-\infty$

What is the connection between infinite limits and vertical asymptotes?

• When a function has an infinite limit at a point, its graph has a vertical asymptote at that value of x

  • This is a vertical line that the graph gets closer and closer to (but never touches or intersects) as x approaches that value

• Vertical asymptotes on the graph of a function are an indication that it has an infinite limit at that x value

  • Conversely, identifying infinite limits for a function lets you identify where the graph of the function has vertical asymptotes

Limits at infinity

What is a limit at infinity?

• Sometimes we are interested in the value of a function as x increases or decreases without bound

  • In such cases we talk about the limit at (positive or negative) infinity of the function

• When considering the behavior of a function f as x increases without bound (i.e. becomes infinitely big in the positive direction) we write $\underset{x\to \infty }{\lim }f\left(x\right)$

• When considering the behavior of a function f as x decreases without bound (i.e. becomes infinitely big in the negative direction) we write $\underset{x\to -\infty }{\lim }f\left(x\right)$

• For some functions, their values as x approaches positive or negative infinity also become unbounded

  • For example, $\underset{x\to \infty }{\lim }\left(x+1\right)=\infty$ and $\underset{x\to -\infty }{\lim }\left(x+1\right)=-\infty$

• But for other functions, their values settle down towards (but never quite reach) a fixed value

  • For example, $\underset{x\to \infty }{\lim }\left(\frac{1}{x}+1\right)=1$ and $\underset{x\to -\infty }{\lim }\left(\frac{1}{x}+1\right)=1$

    • Because $\frac{1}{x}$ gets closer and closer to zero as x becomes large in either the positive or negative directions

• The table below shows the limits at infinity for common functions

What is the connection between limits at infinity and horizontal asymptotes?

• When a function has a finite limit at infinity, its graph has a horizontal asymptote at that value of y

  • This is a horizontal line that the graph gets closer and closer to (but in general never touches or intersects) as x becomes unbounded in the indicated direction

    • For example, if $\underset{x\to \infty }{\lim }f\left(x\right)=3$, then the graph of f will have a horizontal asymptote at y=3

  • The graph becomes 'more and more like' the asymptote as x becomes unbounded

• Horizontal asymptotes on the graph of a function are an indication that the function has a finite limit at infinity

  • Conversely, identifying finite limits at infinity for a function lets you identify where the graph of the function has horizontal asymptotes