use the function and graph to answer the questions about existence, limits, and continuity.\n…

use the function and graph to answer the questions about existence, limits, and continuity.\n f(x)=\begin{cases}-x - 3, & - 5leq xlt - 4\\x + 5, & - 4lt xlt - 2\\-\frac{3}{2}x, & - 2lt xlt0\\0, & 0leq xleq1\\2x - 2, & 1lt xlt3\\2, & x = 3\\-2x + 10, & 3lt xlt5\\1, & x = 5end{cases}\ndoes f(-2) exist?\nyes\nno\ndoes (lim_{x\rightarrow - 2}f(x)) exist?\nno\nyes\ndoes (lim_{x\rightarrow - 2}f(x)) equal f(-2)?\nno\nyes\nis the function continuous at x = - 2?\nyes\nno

use the function and graph to answer the questions about existence, limits, and continuity.\n f(x)=\begin{cases}-x - 3, & - 5leq xlt - 4\\x + 5, & - 4lt xlt - 2\\-\frac{3}{2}x, & - 2lt xlt0\\0, & 0leq xleq1\\2x - 2, & 1lt xlt3\\2, & x = 3\\-2x + 10, & 3lt xlt5\\1, & x = 5end{cases}\ndoes f(-2) exist?\nyes\nno\ndoes (lim_{x\rightarrow - 2}f(x)) exist?\nno\nyes\ndoes (lim_{x\rightarrow - 2}f(x)) equal f(-2)?\nno\nyes\nis the function continuous at x = - 2?\nyes\nno

Answer

Explanation:

Step1: Check existence of f(-2)

The function is not defined for (x = - 2) in any of the sub - intervals of its piece - wise definition. So (f(-2)) does not exist.

Step2: Check existence of (\lim_{x\rightarrow - 2}f(x))

We find the left - hand limit (\lim_{x\rightarrow - 2^{-}}f(x)) and right - hand limit (\lim_{x\rightarrow - 2^{+}}f(x)). For (x\rightarrow - 2^{-}), (f(x)=x + 5), so (\lim_{x\rightarrow - 2^{-}}f(x)=-2 + 5=3). For (x\rightarrow - 2^{+}), (f(x)=-\frac{3}{2}x), so (\lim_{x\rightarrow - 2^{+}}f(x)=-\frac{3}{2}\times(-2)=3). Since (\lim_{x\rightarrow - 2^{-}}f(x)=\lim_{x\rightarrow - 2^{+}}f(x)=3), (\lim_{x\rightarrow - 2}f(x)) exists.

Step3: Compare (\lim_{x\rightarrow - 2}f(x)) and (f(-2))

Since (f(-2)) does not exist and (\lim_{x\rightarrow - 2}f(x)=3), (\lim_{x\rightarrow - 2}f(x)\neq f(-2)).

Step4: Check continuity at (x=-2)

For a function to be continuous at (x = a), (\lim_{x\rightarrow a}f(x)=f(a)). Since (f(-2)) does not exist and (\lim_{x\rightarrow - 2}f(x)) exists, the function is not continuous at (x=-2).

Answer:

Does (f(-2)) exist? No Does (\lim_{x\rightarrow - 2}f(x)) exist? Yes Does (\lim_{x\rightarrow - 2}f(x)) equal (f(-2))? No Is the function continuous at (x = - 2)? No