graph the following function and then find the specified limits. when necessary, state that the limit does…
graph the following function and then find the specified limits. when necessary, state that the limit does not exist.\nf(x)=\begin{cases}x - 1&\text{if }x<2\\1&\text{if }2leq xleq4\\x + 1&\text{if }x>4end{cases}\text{; find }lim_{x\rightarrow2}f(x)\text{ and }lim_{x\rightarrow4}f(x)\nchoose the correct graph below.\na.\nb.\nc.\nd.
Answer
Explanation:
Step1: Find left - hand limit as $x\to2$
For $x < 2$, $f(x)=x - 1$. So, $\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{-}}(x - 1)=2-1 = 1$.
Step2: Find right - hand limit as $x\to2$
For $x\geq2$, when approaching $x = 2$ from the right, $f(x)=1$. So, $\lim_{x\to2^{+}}f(x)=1$. Since $\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{+}}f(x)=1$, $\lim_{x\to2}f(x)=1$.
Step3: Find left - hand limit as $x\to4$
For $x\leq4$, $f(x)=1$. So, $\lim_{x\to4^{-}}f(x)=1$.
Step4: Find right - hand limit as $x\to4$
For $x>4$, $f(x)=x + 1$. So, $\lim_{x\to4^{+}}f(x)=\lim_{x\to4^{+}}(x + 1)=4 + 1=5$. Since $\lim_{x\to4^{-}}f(x)\neq\lim_{x\to4^{+}}f(x)$, $\lim_{x\to4}f(x)$ does not exist.
Answer:
$\lim_{x\to2}f(x)=1$, $\lim_{x\to4}f(x)$ does not exist.