Limits of piece-wise functions exercises

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Here is an opportunity for you to practice finding one- and two-sided limits of piece-wise functions.

Before getting started, you may want to brush-up on what is meant by a piece-wise function and the notation of piece-wise functions. You can do that here.

Let $f(x) = \begin {cases} \frac {x^3 - 8}{x-2} &\text {if $x<1$,} \\ x^3+1 &\text {if $x>1$.} \end {cases}$ Does $\displaystyle \lim _{x \to 2} f(x)$ exist? If it does, give its value. Otherwise write DNE.

$\lim _{x \to 2} f(x) = \answer {9}$

When $x$ is close to 2, what is the rule for $f(x)$ ?

Let $g(x) = \begin {cases} \frac {x^2+5x}{x} & x<0 \\ 1 & 0 \leq x \leq 1 \\ \frac {x}{\sqrt {x-1} +1} & x>1 \end {cases}$ Use $g(x)$ to evaluate the following limits if they exist. Otherwise, write DNE.

$\lim _{x \to -2} g(x) = \answer {3}$ $\lim _{x \to 0^-} g(x) = \answer {5}$ $\lim _{x \to 0^+} g(x) = \answer {1}$ $\lim _{x \to 0} g(x) = \answer {DNE}$ $\lim _{x \to 1^-} g(x) = \answer {1}$ $\lim _{x \to 1^+} g(x) = \answer {1}$ $\lim _{x \to 1} g(x) = \answer {1}$ $\lim _{x \to 5} g(x) = \answer {\frac {5}{3}}$

To evaluate the 2-sided limits, you will first need to evaluate the corresponding 1-sided limits.

The next few problems will involve re-writing absolute value functions as piece-wise functions. If you are unfamiliar with absolute value functions in general or need to review how to re-write absolute value functions as piece-wise functions, take a look at this link.

Let $S(x) = \frac {|x|}{x}$ . Does $\displaystyle \lim _{x \to -4} S(x)$ exist? If it does, give its value. Otherwise write DNE.

$\lim _{x \to -4} S(x) = \answer {-1}$

When $x$ is close to $x=-4$ , what is $|x|$ equal to?

Let $f(t) = \frac {t^2 - 12t +35}{|t-7|}$ . Use $f(t)$ to evaluate the following limits if they exist. Otherwise, write DNE.

$\lim _{x \to 7^-} f(t) = \answer {-2}$
$\lim _{x \to 7^+} f(t) = \answer {2}$
$\lim _{x \to 7} f(t) = \answer {DNE}$

Absolute value functions are piece-wise functions, so you may want to re-write $f(t)$ as an explicit piece-wise function before trying to evaluate the limits.

Let $K(x) = \frac {x^2}{|x|}$ . Use $K(x)$ to evaluate the following limit if it exists. Otherwise, write DNE.

$\lim _{x \to 0} K(x) = \answer {0}$

Let $f(x) = \begin {cases} x^2+55 &\text {if $x<3$,}\\ 0 &\text {if $x=3$,} \\ b^x &\text {if $x>3$.} \end {cases}$ What must the be the value of $b$ to make $\displaystyle \lim _{x \to 3} f(x)$ exist?

$b = \answer {4}$

The left- and right-hand limits at $x=3$ must be equal in order for $\displaystyle \lim _{x \to 3} f(x)$ to exist. Use this information to set up an equation in terms of $b$ , and then solve for $b$ .

Let $f(x) = \begin {cases} |x| &\text {if $x<1$,} \\ \frac {x^2-a^2}{x-a} &\text {if $x>1$.} \end {cases}$ If $\displaystyle \lim _{x \to 1} f(x)$ exists, what is $a$ ?

$a = \answer {0}$