Question

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

Explanation:

Part (a): $\boldsymbol{\lim_{x \to -2^-} F(x)}$

Step1: Analyze left - hand limit

As $x$ approaches $-2$ from the left (values less than $-2$), we look at the graph of the function $y = F(x)$. The left - hand part of the graph (for $x < - 2$) is a line that approaches the point at $x=-2$. From the graph, we can see that as $x$ gets closer to $-2$ from the left, the $y$ - value of the function approaches $0$.

Step1: Analyze right - hand limit

As $x$ approaches $-2$ from the right (values greater than $-2$), we look at the graph of the function $y = F(x)$. The right - hand part of the graph (for $x > - 2$) is a line that approaches the point at $x = - 2$. From the graph, we can see that as $x$ gets closer to $-2$ from the right, the $y$ - value of the function approaches $0$.

Step1: Recall the definition of the limit

The limit $\lim_{x \to a}f(x)$ exists if and only if $\lim_{x \to a^-}f(x)=\lim_{x \to a^+}f(x)$.

Step2: Use results from (a) and (b)

We know from part (a) that $\lim_{x \to -2^-} F(x) = 0$ and from part (b) that $\lim_{x \to -2^+} F(x)=0$. Since the left - hand limit and the right - hand limit are equal, $\lim_{x \to -2} F(x)=0$.

Answer:

$\lim_{x \to -2^-} F(x)=0$

Part (b): $\boldsymbol{\lim_{x \to -2^+} F(x)}$