Question

f(x)=\

$$\begin{cases}17 & x < -3\\\\ -x + 14 & -3 \\leq x < 10\\\\ 2 & x = 10\\\\ 5 & x > 10\\end{cases}$$

evaluate the following limits. it may help to sketch the function. give an exact answer if a limit is a number. otherwise, enter $-\infty$ or $\infty$ if a limit is infinite, or enter dne if a limit does not exist in another way.
\

$$\begin{aligned} \\text{a. } & \\lim_{x \\to 10^{-}} f(x) = \\\\ \\text{b. } & \\lim_{x \\to 10^{+}} f(x) = \\\\ \\text{c. } & \\lim_{x \\to 10} f(x) = \\\\ \\text{d. } & \\lim_{x \\to -3^{-}} f(x) = \\\\ \\text{e. } & \\lim_{x \\to -3^{+}} f(x) = \\\\ \\text{f. } & \\lim_{x \\to -3} f(x) = \\\\ \\end{aligned}$$

Explanation:

Step1: Left limit at x=10

Use $f(x)=-x+14$ for $x<10$
$\lim_{x \to 10^-} f(x) = -10 + 14 = 4$

Step2: Right limit at x=10

Use $f(x)=5$ for $x>10$
$\lim_{x \to 10^+} f(x) = 5$

Step3: Overall limit at x=10

Left ≠ right limit, so DNE

Step4: Left limit at x=-3

Use $f(x)=17$ for $x<-3$
$\lim_{x \to -3^-} f(x) = 17$

Step5: Right limit at x=-3

Use $f(x)=-x+14$ for $x>-3$
$\lim_{x \to -3^+} f(x) = -(-3) + 14 = 17$

Step6: Overall limit at x=-3

Left = right limit, so 17

Answer:

a. $4$
b. $5$
c. DNE
d. $17$
e. $17$
f. $17$