Question

1. using the graph, find each limit or explain why the limit does not exist.

a. $\lim_{x\to - 1}f(x)$
b. $\lim_{x\to 1}f(x)$
c. $\lim_{x\to 0^{+}}f(x)$
d. $\lim_{x\to 2}f(x)$

Explanation:

Step1: Analyze $\lim_{x

ightarrow - 1}f(x)$
As \(x\) approaches \(-1\), we look at the behavior of the function near \(x = - 1\). The left - hand limit and the right - hand limit (since there is no break in the behavior near \(x=-1\) from the graph) approach the same value.

Step2: Analyze $\lim_{x

ightarrow1}f(x)$
For \(\lim_{x
ightarrow1}f(x)\), we consider the left - hand limit and the right - hand limit. As \(x\) approaches \(1\) from the left (\(x\to1^{-}\)), the function approaches \(1\), and as \(x\) approaches \(1\) from the right (\(x\to1^{+}\)), the function approaches \(1\).

Step3: Analyze $\lim_{x

ightarrow0^{+}}f(x)$
When \(x\) approaches \(0\) from the positive side (\(x\to0^{+}\)), we observe the graph. The function approaches \(0\) as \(x\) gets closer to \(0\) from the right.

Step4: Analyze $\lim_{x

ightarrow2}f(x)$
For \(\lim_{x
ightarrow2}f(x)\), we check the left - hand limit and the right - hand limit. As \(x\) approaches \(2\) from the left (\(x\to2^{-}\)), the function approaches \(1\), and as \(x\) approaches \(2\) from the right (\(x\to2^{+}\)), the function approaches \(1\).

Answer:

a. \(\lim_{x
ightarrow - 1}f(x)=1\)
b. \(\lim_{x
ightarrow1}f(x)=1\)
c. \(\lim_{x
ightarrow0^{+}}f(x)=0\)
d. \(\lim_{x
ightarrow2}f(x)=1\)