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1.2E: Exercises

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Exercise $$\PageIndex{1}$$

In exercises 1 - 4, consider the graph of the function$$y=f(x)$$ shown here. Which of the statements about $$y=f(x)$$ are true and which are false? Explain why a statement is false.

1) $$\displaystyle \lim_{x→10}f(x)=0$$

2) $$\displaystyle \lim_{x→−2^+}f(x)=3$$

False; $$\displaystyle \lim_{x→−2^+}f(x)=+∞$$

3) $$\displaystyle \lim_{x→−8}f(x)=f(−8)$$

4) $$\displaystyle \lim_{x→6}f(x)=5$$

False; $$\displaystyle \lim_{x→6}f(x)$$ DNE since $$\displaystyle \lim_{x→6^−}f(x)=2$$ and $$\displaystyle \lim_{x→6^+}f(x)=5$$.

Exercise $$\PageIndex{2}$$

In exercises 1 - 5, use the following graph of the function $$y=f(x)$$ to find the values, if possible. Estimate when necessary.

1) $$\displaystyle \lim_{x→1^−}f(x)$$

2) $$\displaystyle \lim_{x→1^+}f(x)$$

$$2$$

3) $$\displaystyle \lim_{x→1}f(x)$$

4) $$\displaystyle \lim_{x→2}f(x)$$

$$1$$

5) $$f(1)$$

Exercise $$\PageIndex{3}$$

In exercises 1 - 4, use the graph of the function $$y=f(x)$$ shown here to find the values, if possible. Estimate when necessary.

1) $$\displaystyle \lim_{x→0^−}f(x)$$

$$1$$

2) $$\displaystyle \lim_{x→0^+}f(x)$$

3) $$\displaystyle \lim_{x→0}f(x)$$

DNE

4) $$\displaystyle \lim_{x→2}f(x)$$

Exercise $$\PageIndex{4}$$

In exercises 1 - 6, use the graph of the function $$y=f(x)$$ shown here to find the values, if possible. Estimate when necessary.

1) $$\displaystyle \lim_{x→−2^−}f(x)$$

$$0$$

2) $$\displaystyle \lim_{x→−2^+}f(x)$$

3) $$\displaystyle \lim_{x→−2}f(x)$$

DNE

4) $$\displaystyle \lim_{x→2^−}f(x)$$

5) $$\displaystyle \lim_{x→2^+}f(x)$$

$$2$$

6) $$\displaystyle \lim_{x→2}f(x)$$

Exercise $$\PageIndex{5}$$

1. In exercises a-c, use the graph of the function $$y=g(x)$$ shown here to find the values, if possible. Estimate when necessary.

a) $$\displaystyle \lim_{x→0^−}g(x)$$

$$3$$

b) $$\displaystyle \lim_{x→0^+}g(x)$$

c) $$\displaystyle \lim_{x→0}g(x)$$

DNE

2. In exercises a-c, use the graph of the function $$y=h(x)$$ shown here to find the values, if possible. Estimate when necessary.

a) $$\displaystyle \lim_{x→0^−}h(x)$$

b) $$\displaystyle \lim_{x→0^+}h(x)$$

$$0$$

c) $$\displaystyle \lim_{x→0}h(x)$$

Exercise $$\PageIndex{6}$$

In exercises a-e, use the graph of the function $$y=f(x)$$ shown here to find the values, if possible. Estimate when necessary.

a) $$\displaystyle \lim_{x→0^−}f(x)$$

$$-2$$

b) $$\displaystyle \lim_{x→0^+}f(x)$$

c) $$\displaystyle \lim_{x→0}f(x)$$

DNE

d) $$\displaystyle \lim_{x→1}f(x)$$

e) $$\displaystyle \lim_{x→2}f(x)$$

$$0$$

Exercise $$\PageIndex{7}$$

In exercises 1 - 5, sketch the graph of a function with the given properties.

1) $$\displaystyle \lim_{x→2}f(x)=1, \quad \lim_{x→4^−}f(x)=3, \quad \lim_{x→4^+}f(x)=6, \quad x=4$$ is not defined.

2) $$\displaystyle \lim_{x→−∞}f(x)=0, \quad \lim_{x→−1^−}f(x)=−∞, \quad \lim_{x→−1^+}f(x)=∞,\quad \lim_{x→0}f(x)=f(0), \quad f(0)=1, \quad \lim_{x→∞}f(x)=−∞$$

3) $$\displaystyle \lim_{x→−∞}f(x)=2, \quad \lim_{x→3^−}f(x)=−∞, \quad \lim_{x→3^+}f(x)=∞, \quad \lim_{x→∞}f(x)=2, \quad f(0)=-\frac{1}{3}$$

4) $$\displaystyle \lim_{x→−∞}f(x)=2,\quad \lim_{x→−2}f(x)=−∞,\quad \lim_{x→∞}f(x)=2,\quad f(0)=0$$

5) $$\displaystyle \lim_{x→−∞}f(x)=0,\quad \lim_{x→−1^−}f(x)=∞,\quad \lim_{x→−1^+}f(x)=−∞, \quad f(0)=−1, \quad \lim_{x→1^−}f(x)=−∞, \quad \lim_{x→1^+}f(x)=∞, \quad \lim_{x→∞}f(x)=0$$

Exercise $$\PageIndex{8}$$

1) Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, $$x$$, is shown here. We are mainly interested in the location of the front of the shock, labeled $$X_{SF}$$ in the diagram.

a. Evaluate $$\displaystyle \lim_{x→X_{SF}^+}ρ(x)$$.

b. Evaluate $$\displaystyle \lim_{x→X_{SF}^−}ρ(x)$$.

c. Evaluate $$\displaystyle \lim_{x→X_{SF}}ρ(x)$$. Explain the physical meanings behind your answers.

a. $$ρ_2$$ b. $$ρ_1$$ c. DNE unless $$ρ_1=ρ_2$$. As you approach $$X_{SF}$$ from the right, you are in the high-density area of the shock. When you approach from the left, you have not experienced the “shock” yet and are at a lower density.

2) A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where $$x$$ is the position in meters of the runner and $$t$$ is time in seconds. What is $$\displaystyle \lim_{t→2}x(t)$$? What does it mean physically?

$$t(sec)$$ $$x(m)$$
1.75 4.5
1.95 6.1
1.99 6.42
2.01 6.58
2.05 6.9
2.25 8.5

Exercise $$\PageIndex{9}$$ 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 $$x=5$$.

Under construction.

Exercise $$\PageIndex{10}$$: Finding limits using Graphs

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

1.

(a)2,(b)2,(c) 2,(d) 1, (e)DNE, (f)1

2.

Under construction.

3.

Under construction.

4.

Under construction.

Under construction.

5..

Under construction.

6.

Under construction.

7.

Under construction.

Exercise $$\PageIndex{11}$$: 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) $$\lim\limits_{x\to \infty}f(x)$$

Under construction.

Exercise $$\PageIndex{11}$$: 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.

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 $$\PageIndex{12}$$: Infinite limits

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

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