Question

1. for the function f graphed in the accompanying figure, find (a) $limlimits_{x \to 3^-} f(x)$ (b) $limlimits_{x \to 3^+} f(x)$ (c) $limlimits_{x \to 3} f(x)$ (d) $f(3)$ (e) $limlimits_{x \to -\infty} f(x)$ (f) $limlimits_{x \to +\infty} f(x)$.

Explanation:

Part (a)

Step1: Analyze left - hand limit as \(x

ightarrow3^{-}\)
As \(x\) approaches \(3\) from the left (values less than \(3\)), we look at the graph of \(y = f(x)\). The left - hand side of \(x = 3\) (for \(x<3\)) has a horizontal line. From the graph, we can see that as \(x\) gets closer to \(3\) from the left, the \(y\) - value approaches \(3\). So, \(\lim_{x
ightarrow3^{-}}f(x)=3\).

Part (b)

Step1: Analyze right - hand limit as \(x

ightarrow3^{+}\)
As \(x\) approaches \(3\) from the right (values greater than \(3\)), we look at the graph of \(y = f(x)\). The right - hand side of \(x = 3\) (for \(x > 3\)) has a horizontal line. From the graph, we can see that as \(x\) gets closer to \(3\) from the right, the \(y\) - value approaches \(3\). So, \(\lim_{x
ightarrow3^{+}}f(x)=3\).

Part (c)

Step1: Recall the definition of 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 \(\lim_{x
ightarrow3^{-}}f(x) = 3\) and \(\lim_{x
ightarrow3^{+}}f(x)=3\). Since the left - hand limit and the right - hand limit are equal, \(\lim_{x
ightarrow3}f(x)=3\).

Part (d)

Answer:

s:
(a) \(\boldsymbol{\lim_{x
ightarrow3^{-}}f(x)=3}\)

(b) \(\boldsymbol{\lim_{x
ightarrow3^{+}}f(x)=3}\)

(c) \(\boldsymbol{\lim_{x
ightarrow3}f(x)=3}\)

(d) \(\boldsymbol{f(3) = 1}\)

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

(f) \(\boldsymbol{\lim_{x
ightarrow+\infty}f(x)=3}\)