Question

1. for the function ( f ) graphed in the accompanying figure, find

(a) ( limlimits_{x \to 0^-} f(x) )
(b) ( limlimits_{x \to 0^+} f(x) )
(c) ( limlimits_{x \to 0} f(x) )
(d) ( f(0) )
(e) ( limlimits_{x \to -infty} f(x) )
(f) ( limlimits_{x \to +infty} f(x) ).
figure ex-10 (graph of ( y = f(x) ))

Explanation:

Part (a): $\boldsymbol{\lim_{x \to 0^-} f(x)}$

Step1: Analyze left - hand limit

As $x$ approaches $0$ from the left (values of $x$ less than $0$), we look at the graph of the function for $x<0$. The left - hand part of the graph (for $x < 0$) approaches a $y$ - value. From the graph, when $x$ approaches $0$ from the left, the function approaches $1$ (by looking at the open circle and the trend of the left - hand curve).

Step1: Analyze right - hand limit

As $x$ approaches $0$ from the right (values of $x$ greater than $0$), we look at the graph of the function for $x > 0$. The right - hand part of the graph (for $x>0$) near $x = 0$ has a trend. From the graph, when $x$ approaches $0$ from the right, the function goes to $-\infty$ (since the graph has a vertical asymptote - like behavior as $x$ approaches $0$ from the right, going downwards).

Step1: Recall limit existence condition

For the limit $\lim_{x\to a}f(x)$ to exist, $\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)$. Here, $a = 0$, $\lim_{x\to 0^-}f(x) = 1$ and $\lim_{x\to 0^+}f(x)=-\infty$. Since $1
eq-\infty$, the two - sided limit does not exist.

Answer:

$\lim_{x \to 0^-} f(x)=1$

Part (b): $\boldsymbol{\lim_{x \to 0^+} f(x)}$