How to Estimate Limit Values with Graphs

Quick Summary

  1. When working with graphs, the best we can do is estimate the value of limits.
  2. If the one-sided limits seem to be equal, we use their value as the value of the limit.

Examples

Example 1

Use the graph to estimate $$\displaystyle\lim\limits_{x\to4} f(x)$$

Step 1

Examine the limit from the left.

Step 2

Examine the limit from the right.

Step 3

The one-sided limits are the same, so the limit exists.

Answer: $$\displaystyle\lim\limits_{x\to4} f(x) \approx 5$$

Example 2
Step 1

Use the graph to estimate $$\displaystyle\lim\limits_{x\to-3} f(x)$$

Step 1

Examine the limit from the left.

Step 2

Examine the limit from the right.

Step 3

The one-sided limits are the same, so the limit exists.

Answer: $$\displaystyle\lim\limits_{x\to-3}f(x) \approx 2$$

Example 3
Step 1

Use the graph to evaluate $$\displaystyle\lim\limits_{x\to0} f(x)$$

Step 1

Examine the limit from the left and from the right.

Step 2

Examine the one-sided limits.

The limit from the left is not the same as the limit from the right.

Answer: The limit does not exist.

Practice Problems

Use the graph below to estimate the value of the limits in questions 1--5.

Problem 1

$$\displaystyle\lim\limits_{x\to-3} f(x) =$$

Answer: $$\displaystyle\lim\limits_{x\to-3} f(x) \approx 4$$

Problem 2

$$\displaystyle\lim\limits_{x\to-2} f(x) =\red ?$$

Reminder of this graph

Answer: $$\displaystyle\lim\limits_{x\to-2} f(x)$$ does not exist.

Problem 3

$$\displaystyle\lim\limits_{x\to0} f(x) =\red ?$$

Reminder of this graph

Answer: $$\displaystyle\lim\limits_{x\to0} f(x) \approx 0$$

Problem 4

$$\displaystyle\lim\limits_{x\to2} f(x) =\red ?$$

Reminder of this graph

Answer: $$\displaystyle\lim\limits_{x\to2} f(x) \approx 1$$

Problem 5

$$\displaystyle\lim\limits_{x\to4} f(x) = \red{?}$$

Reminder of this graph

The limit from the right doesn't exist.

Answer: $$\displaystyle\lim\limits_{x\to4} f(x)$$ does not exist.

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