Quick Summary
- When working with graphs, the best we can do is estimate the value of limits.
- 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 $$\displaystyle\lim\limits_{x\to4} f(x)$$
![](images/limit-graph-example-1.png)
Examine the limit from the left.
![](images/limit-from-left-example-1.png)
Examine the limit from the right.
![](images/limit-from-right-example-1.png)
The one-sided limits are the same, so the limit exists.
Answer: $$\displaystyle\lim\limits_{x\to4} f(x) \approx 5$$
Example 2
Use the graph to estimate $$\displaystyle\lim\limits_{x\to-3} f(x)$$
![](images/limit-asymptote-example-2.png)
Examine the limit from the left.
![](images/limit-from-left-asymptote-example-2.png)
Examine the limit from the right.
![](images/limit-from-right-asymptote-example-2.png)
The one-sided limits are the same, so the limit exists.
Answer: $$\displaystyle\lim\limits_{x\to-3}f(x) \approx 2$$
Example 3
Use the graph to evaluate $$\displaystyle\lim\limits_{x\to0} f(x)$$
![](images/limit-piecewise-example-3.png)
Examine the limit from the left and from the right.
![](images/limit-piecewise-answer-example-3.png)
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.
![](images/limit-graph-for-questions.png)
![](images/limit-solution-question-1.png)
Answer: $$\displaystyle\lim\limits_{x\to-3} f(x) \approx 4$$
![](images/limit-solution-question-2.png)
Answer: $$\displaystyle\lim\limits_{x\to-2} f(x)$$ does not exist.
![](images/limit-solution-question-3.png)
Answer: $$\displaystyle\lim\limits_{x\to0} f(x) \approx 0$$
![](images/limit-solution-question-4.png)
Answer: $$\displaystyle\lim\limits_{x\to2} f(x) \approx 1$$
![](images/limit-solution-question-5.png)
The limit from the right doesn't exist.
Answer: $$\displaystyle\lim\limits_{x\to4} f(x)$$ does not exist.