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# 1.2E Exercises

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### Exercise $$\PageIndex{1}$$ Terms and Concepts

1.  What are the  ways in which a limit may fail to exist?
2.  T/F: If $$\lim\limits_{x\to1-}f(x)=5$$, then $$\lim\limits_{x\to1}f(x)=5$$
3.  T/F: If $$\lim\limits_{x\to1-}f(x)=5$$, then $$\lim\limits_{x\to1+}f(x)=5$$
4.  T/F: If $$\lim\limits_{x\to1}f(x)=5$$, then $$\lim\limits_{x\to1-}f(x)=5$$
5. T/F: If $$\lim\limits_{x\to 1^-}f(x)=-\infty$$, then $$\lim\limits_{x\to 1^+}f(x)=\infty$$.
6.  T/F: If $$\lim\limits_{x\to 5}f(x)=\infty$$, then $$f$$ has a vertical asymptote at $$x=5$$.

Answer

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### Exercise $$\PageIndex{2}$$: Finding limits using Graphs

Evaluate each expression using the given graph of $$f(x)$$.

Answer

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### Exercise $$\PageIndex{3}$$: Finding limits using Graphs

Evaluate each expression using the given graph of $$f(x)$$.

Answer

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### Exercise $$\PageIndex{4}$$: Finding limits using Graphs

Answer

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### Exercise $$\PageIndex{5}$$: Finding limits using Graphs

Answer

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### Exercise $$\PageIndex{6}$$: Finding limits using Graphs

Answer

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### Exercise $$\PageIndex{7}$$: Finding limits using Graphs

Answer

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### Exercise $$\PageIndex{8}$$: Finding limits using Graphs

Answer

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### Exercise $$\PageIndex{9}$$: Infinite limits

In Exercises 1-6, evaluate the given limits using the graph of the function.

1. $$f(x) = \frac{1}{(x+1)^2}$$

(a) $$\lim\limits_{x\to -1^-}f(x)$$

(b) $$\lim\limits_{x\to -1^+}f(x)$$

2. $$f(x) = \frac{1}{(x-3)(x-5)^2}$$

(a) $$\lim\limits_{x\to 3^-}f(x)$$

(b) $$\lim\limits_{x\to 3^+}f(x)$$

(c) $$\lim\limits_{x\to 3}f(x)$$

(d) $$\lim\limits_{x\to 5^-}f(x)$$

(e) $$\lim\limits_{x\to 5^+}f(x)$$

(f) $$\lim\limits_{x\to 5}f(x)$$

3. $$f(x) = \frac{1}{e^x+1}$$

(a) $$\lim\limits_{x\to -\infty}f(x)$$

(b) $$\lim\limits_{x\to \infty}f(x)$$

(c) $$\lim\limits_{x\to 0^-}f(x)$$

(d) $$\lim\limits_{x\to 0^+}f(x)$$

4. $$f(x) = x^2\sin (\pi x)$$

(a) $$\lim\limits_{x\to -\infty}f(x)$$

(b) $$\lim\limits_{x\to \infty}f(x)$$

5. $$f(x)=\cos (x)$$

(a) $$\lim\limits_{x\to -\infty}f(x)$$

(b) $$\lim\limits_{x\to \infty}f(x)$$

6. $$f(x) = 2^x +10$$

(a) $$\lim\limits_{x\to -\infty}f(x)$$

(b) $$\lim\limits_{x\to \infty}f(x)$$

Answer

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### Exercise $$\PageIndex{10}$$: One sided limits

Given $$\displaystyle f(x)= \left\{ \begin{array}{ccc} x^3+1 & \mbox{ if } x <0 \\ 0 & \mbox{ if } x =0 \\ \sqrt{x+1}-2 & \mbox{ if } x >0 \\ \end{array} \right.$$

Identify each limit:

1. $$\displaystyle \lim_{x \to 0^-} f(x)$$
2. $$\displaystyle \lim_{x \to 0^+} f(x)$$
3. $$\displaystyle \lim_{x \to 0} f(x)$$
4. $$\displaystyle \lim_{x \to -1} f(x)$$
5. $$\displaystyle \lim_{x \to 3} f(x)$$
Hint:

Draw a graph.

Answer:

11, -1, DNE, 0, 0

Solution:
1. $$\displaystyle \lim_{x \to 0^-} f(x)=0^3+11$$
2. $$\displaystyle \lim_{x \to 0^+} f(x)= \sqrt{0+1}-2=-1$$
3. Since $$\displaystyle \lim_{x \to 0^-} f(x) \ne \displaystyle \lim_{x \to 0^+} f(x)$$, $$\displaystyle \lim_{x \to 0} f(x) =$$  DNE. \
4. $$\displaystyle \lim_{x \to -1} f(x)=(-1)^3+1=0$$
5. $$\displaystyle \lim_{x \to 3} f(x)= \sqrt{3+1}-2=0$$

### Exercise $$\PageIndex{11}$$: Infinite limits

$$\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36}$$

Answer

Since $$\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36}= \frac{12}{0}$$, and $$\frac{5.9+6}{(5.9)^2-36}<0,$$

$$\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36} =-\infty$$.

## Contributors

Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/