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

Revision Note

Author

Roger B

Expertise

Maths

Infinite limits

What do we mean by 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 $\lim_{x \to c} f (x) = \infty$ .
- For example, $\lim_{x \to 0} \frac{1}{x^{2}} = \infty$
If the value of a function f decreases without bound as x approaches some value c, then we write $\lim_{x \to c} f (x) = - \infty$ .
- For example, $\lim_{x \to 0} (- \frac{1}{x^{2}}) = - \infty$
It is possible for the one-sided limits to be different
- For example, $\lim_{x \to 0^{+}} \frac{1}{x} = \infty$
- But $\lim_{x \to 0^{-}} \frac{1}{x} = - \infty$

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

Worked Example

The figure below shows the graph of the function $f$ defined by $f (x) = \frac{1}{{(x - 2)}^{2}}$ . The dashed line is a vertical asymptote of the graph.

Graph of y=1/(x-2)^2, including vertical asymptote at x=2

What is $\lim_{x \to 2} f (x)$ ?

Answer:

f(x) is positive for all values of x except 2

As x approaches 2, the denominator gets closer and closer to zero, and the value of the function increases without bound (i.e. gets bigger and bigger in the positive direction)

$\lim_{x \to 2} f (x) = \infty$

Limits at infinity

What do we mean by 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 $\lim_{x \to \infty} f (x)$
When considering the behavior of a function f as x decreases without bound (i.e. becomes infinitely big in the negative direction) we write $\lim_{x \to - \infty} f (x)$
For some functions, their values as x approaches positive or negative infinity also become unbounded
- For example, $\lim_{x \to \infty} (x + 1) = \infty$ and $\lim_{x \to - \infty} (x + 1) = - \infty$
But for other functions, their values settle down towards (but never quite reach) a fixed value
- For example, $\lim_{x \to \infty} (\frac{1}{x} + 1) = 1$ and $\lim_{x \to - \infty} (\frac{1}{x} + 1) = 1$
  - Because $\frac{1}{x}$ gets closer and closer to zero as x becomes in large in either the positive or negative directions

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 $\lim_{x \to \infty} f (x) = 3$ , 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

Exam Tip

If an exam question is using a function to model a real-world scenario, be sure to interpret any limits at infinity in the context of the question.

Worked Example

The figure below shows the graph of the function $P$ defined by $P (t) = 4 - \frac{3}{t + 1}, t \geq 0$ .

$P$ is being used to model the population (in hundreds) of squirrels in a particular area of woodland at time $t$ years after the beginning of a study.

(a) Find $\lim_{t \to \infty} P (t)$ .

Answer:

As t gets bigger and bigger in the positive direction, the fraction $\frac{3}{t + 1}$ will become closer and closer to zero

Therefore the function will get closer and closer to 4 (without ever quite reaching 4)

$\lim_{t \to \infty} P (t) = 4$

(b) Interpret your answer for part (a) in the context of this problem.

Answer:

Connect the limit in part (a) to the context given in the question

Over time the population of squirrels in the woodland will approach 400.

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

Revision Note

Infinite limits

What do we mean by an infinite limit?

What is the connection between infinite limits and vertical asymptotes?

Worked Example

Limits at infinity

What do we mean by a limit at infinity?

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

Exam Tip

Worked Example

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Author: Roger B