Topic 8 of 8

Informal limits

Step 14 · Trigonometry and precalculus topics

Understand a limit as the value a function approaches, find limits by simplifying, from tables and from graphs, check one-sided limits, and find limits as x grows without bound.

Approaching a value

A limit describes the value a function gets close to as x gets close to some number, whether or not the function actually reaches it there.

\lim_{x \to a} f(x) = L

This reads: as x approaches a, f(x) approaches L.

Example: Closer and closer

Find $\lim_{x \to 2} (3x + 1)$ .

As x gets close to 2, 3x + 1 gets close to 7.
Here you can simply put x = 2 in.

Answer7

Example: A hole

Find $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ .

At x = 3 the fraction is 0 ÷ 0, which is undefined.
For every other x it equals x + 3, which approaches 6.

Answer6

Limits from tables and graphs

A table of values close to a, from both sides, or a graph shows where a function is heading.

x	2.9	2.99	2.999	3.001	3.01	3.1
f(x)	5.9	5.99	5.999	6.001	6.01	6.1

Values of (x² − 9) ÷ (x − 3) close to x = 3, from the left and from the right: both approach 6.

Example: sin x over x

Estimate $\lim_{x \to 0} \frac{\sin x}{x}$ , with x in radians.

x = 0.1 gives about 0.998, and x = 0.01 gives about 0.99998.
The negative side gives the same values.

Answer1

Example: One-sided limits

f(x) = 1 for x < 0 and f(x) = 3 for x ≥ 0. What happens as x approaches 0?

From the left, f approaches 1.
From the right, f approaches 3.

AnswerThe two sides differ, so the limit does not exist

Limits at infinity

A limit at infinity describes what f(x) approaches as x grows without bound. It is the horizontal asymptote.

Example: A rational function

Find $\lim_{x \to \infty} \frac{2x + 3}{x - 1}$ .

For huge x the leading terms dominate: 2x ÷ x = 2.
At x = 1,000: 2,003 ÷ 999 is about 2.005.

Answer2

Example: Shrinking to zero

Find $\lim_{x \to \infty} \frac{5}{x^{2}}$ .

The bottom grows without bound, so the fraction shrinks toward 0.

Answer0

Example: Decay

Find $\lim_{x \to \infty} 4\left(\tfrac{1}{2}\right)^{x}$ .

Each step halves the value: 2, 1, 0.5, and so on.
It gets as close to 0 as you like.