Estimating Limit Values with Graphs

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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