Question

1. for the function f graphed in the accompanying figure, find (a) $limlimits_{x \to 2^-} f(x)$ (b) $limlimits_{x \to 2^+} f(x)$ (c) $limlimits_{x \to 2} f(x)$ (d) $f(2)$ (e) $limlimits_{x \to -\infty} f(x)$ (f) $limlimits_{x \to +\infty} f(x)$. figure ex-2

Explanation:

Part (a)

Step1: Analyze left - hand limit as \(x

ightarrow2^{-}\)
To find \(\lim_{x
ightarrow2^{-}}f(x)\), we look at the behavior of the function \(f(x)\) as \(x\) approaches \(2\) from the left - hand side (values of \(x\) less than \(2\)). From the graph, as \(x\) gets closer to \(2\) from the left, the \(y\) - value of the function approaches \(2\).
So, \(\lim_{x
ightarrow2^{-}}f(x)=2\)

Part (b)

Step1: Analyze right - hand limit as \(x

ightarrow2^{+}\)
To find \(\lim_{x
ightarrow2^{+}}f(x)\), we look at the behavior of the function \(f(x)\) as \(x\) approaches \(2\) from the right - hand side (values of \(x\) greater than \(2\)). From the graph, as \(x\) gets closer to \(2\) from the right, the \(y\) - value of the function approaches \(0\) (since there is an open circle at \((2,0)\) on the right - hand part of the graph near \(x = 2\)).
So, \(\lim_{x
ightarrow2^{+}}f(x)=0\)

Part (c)

Step1: Recall the definition of the two - sided limit

The two - sided limit \(\lim_{x
ightarrow a}f(x)\) exists if and only if \(\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)\). We found that \(\lim_{x
ightarrow2^{-}}f(x) = 2\) and \(\lim_{x
ightarrow2^{+}}f(x)=0\). Since \(2
eq0\), the two - sided limit \(\lim_{x
ightarrow2}f(x)\) does not exist.

Part (d)

Answer:

Step1: Analyze the limit as \(x

ightarrow+\infty\)
To find \(\lim_{x
ightarrow+\infty}f(x)\), we look at the behavior of the function as \(x\) becomes very large in the positive direction (moves to the right - most part of the graph). From the graph, as \(x
ightarrow+\infty\), the function \(f(x)\) approaches \(2\) (the horizontal line on the right - hand side of the graph is at \(y = 2\)).
So, \(\lim_{x
ightarrow+\infty}f(x)=2\)

Final Answers:

(a) \(\boldsymbol{\lim_{x
ightarrow2^{-}}f(x)=2}\)

(b) \(\boldsymbol{\lim_{x
ightarrow2^{+}}f(x)=0}\)

(c) \(\boldsymbol{\lim_{x
ightarrow2}f(x)\text{ does not exist}}\)

(d) \(\boldsymbol{f(2) = 2}\)

(e) \(\boldsymbol{\lim_{x
ightarrow-\infty}f(x)=0}\)

(f) \(\boldsymbol{\lim_{x
ightarrow+\infty}f(x)=2}\)