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University of California, Berkeley

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Problem 31

In Exercises 31 and $32,$ sketch the graph of $f$ . Then identify the values of $c$ for which $$\lim _{x \rightarrow e} f(x)$$

$$f(x)=\left\{\begin{array}{ll}{x^{2},} & {x \leq 2} \\ {8-2 x,} & {2 < x < 4} \\ {4,} & {x \geq 4}\end{array}\right.$$

Answer

See solution for explanation

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## Discussion

## Video Transcript

Okay, so it's practical function. See, that with X is less than or equal to our function is X squared. So we know X squared has ordinates. What value did it at two squares? This one I can and then it goes off. Okay. And the next we have eight minutes to act that has an accent is up to ask. For what? We're not gonna include that point. Since we're not included, it's gonna look something like this downward line and made another at X rated very quick before we just have four. So we have a doctor here. This is including four. And I was just going off like that. Okay, so then what value of C makes it so our limits LTD. This is so when the limit as explosions sea of our function. And this is one. Let's see, So every point exists is, except for when we're approaching four. Cracked. Whose only approach from the left hand side. We're looking, though, doesn't the right hand side was looking for. This is when she is not equal to four

## Recommended Questions

In Exercises 31 and $32,$ sketch the graph of $f$ . Then identify the values of $c$ for which $$\lim _{x \rightarrow e} f(x)$$

$$f(x)=\left\{\begin{array}{ll}{\sin x,} & {x < 0} \\ {1-\cos x,} & {0 \leq x<\pi} \\ {\cos x,} & {x>\pi}\end{array}\right.$$

Sketching a Graph of a Function In Exercises $31-38,$ sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph.

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Sketching a Graph of a Function In Exercises $31-38,$ sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph.

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Refer to Exercise $32 .$ Find the value of $x$ for each value of $f(x) .$ See Example $5(\mathrm{c})$

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(c) $f(x)=0$

In Exercises $31-34,$ graph the piecewise-defined functions.

$$f(x)=\left\{\begin{array}{ll}{x^{2},} & {x<0} \\ {x^{3},} & {0 \leq x \leq 1} \\ {2 x-1,} & {x>1}\end{array}\right.$4

Sketch the graph of $f(2 x)$ and $f\left(\frac{1}{2} x\right),$ where $f(x)=|x|+1$ (Figure 28 ).

In Exercises $29-34,$ graph the function and identify intervals on which the function is increasing, decreasing, or constant.

$$f(x)=x^{3}-x^{2}-2 x$$

Sketch the graph of each function. Decide whether each function is one-to-one. See Sections 3.2 and 9.2

$$f(x)=x^{2}+3$$