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) $limlimits_{x \to +\infty} f(x)$ figure ex-6

Explanation:

Part (a)

Step1: Analyze left - hand limit at \(x = 3\)

To find \(\lim_{x
ightarrow3^{-}}F(x)\), we look at the values of \(F(x)\) as \(x\) approaches \(3\) from the left side (values less than \(3\)). From the graph, the function approaches the \(y\) - value of the open circle at \(x = 3\) from the left. The open circle at \(x = 3\) has a \(y\) - coordinate of \(2\). So as \(x\) gets closer to \(3\) from the left, \(F(x)\) approaches \(2\).
\(\lim_{x
ightarrow3^{-}}F(x)=2\)

Part (b)

Step1: Analyze right - hand limit at \(x = 3\)

To find \(\lim_{x
ightarrow3^{+}}F(x)\), we look at the values of \(F(x)\) as \(x\) approaches \(3\) from the right side (values greater than \(3\)). From the graph, the function is a straight line, and as \(x\) approaches \(3\) from the right, it also approaches the \(y\) - value of the open circle (since the line passes through the open circle's \(x\) - value and follows the same trend). So \(\lim_{x
ightarrow3^{+}}F(x)=2\)

Part (c)

Step1: Recall the limit existence condition

A 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 \(\lim_{x
ightarrow3^{-}}F(x) = 2\) and \(\lim_{x
ightarrow3^{+}}F(x)=2\). Since the left - hand limit and the right - hand limit are equal, \(\lim_{x
ightarrow3}F(x)=2\)

Part (d)

Answer:

Step1: Analyze the limit as \(x

ightarrow+\infty\)
As \(x\) approaches \(+\infty\), we look at the end - behavior of the function. The function is a straight line with a positive slope. As \(x\) becomes very large in the positive direction (\(x
ightarrow+\infty\)), the \(y\) - value of the function goes to \(+\infty\) (because the line is increasing, so as \(x\) increases without bound, \(y\) increases without bound). So \(\lim_{x
ightarrow+\infty}F(x)=+\infty\)

Final Answers:

(a) \(\boldsymbol{2}\)

(b) \(\boldsymbol{2}\)

(c) \(\boldsymbol{2}\)

(d) \(\boldsymbol{3}\)

(e) \(\boldsymbol{-\infty}\)

(f) \(\boldsymbol{+\infty}\)