Question

1. the graph of f in the figure has vertical asymptotes at x = 1 and x = 2. analyze the following limits. a. $lim_{x

ightarrow1^{-}}f(x)$ b. $lim_{x
ightarrow1^{+}}f(x)$ c. $lim_{x
ightarrow1}f(x)$ d. $lim_{x
ightarrow2^{-}}f(x)$ e. $lim_{x
ightarrow2^{+}}f(x)$ f. $lim_{x
ightarrow2}f(x)$

Explanation:

Step1: Analyze left - hand limit as $x\to1$

As $x$ approaches $1$ from the left side ($x\to1^{-}$), the function $y = f(x)$ goes to $+\infty$. So, $\lim_{x\to1^{-}}f(x)=+\infty$.

Step2: Analyze right - hand limit as $x\to1$

As $x$ approaches $1$ from the right side ($x\to1^{+}$), the function $y = f(x)$ goes to $-\infty$. So, $\lim_{x\to1^{+}}f(x)=-\infty$.

Step3: Analyze limit as $x\to1$

Since $\lim_{x\to1^{-}}f(x)
eq\lim_{x\to1^{+}}f(x)$, $\lim_{x\to1}f(x)$ does not exist.

Step4: Analyze left - hand limit as $x\to2$

As $x$ approaches $2$ from the left side ($x\to2^{-}$), the function $y = f(x)$ goes to $-\infty$. So, $\lim_{x\to2^{-}}f(x)=-\infty$.

Step5: Analyze right - hand limit as $x\to2$

As $x$ approaches $2$ from the right side ($x\to2^{+}$), the function $y = f(x)$ goes to $+\infty$. So, $\lim_{x\to2^{+}}f(x)=+\infty$.

Step6: Analyze limit as $x\to2$

Since $\lim_{x\to2^{-}}f(x)
eq\lim_{x\to2^{+}}f(x)$, $\lim_{x\to2}f(x)$ does not exist.

Answer:

a. $\lim_{x\to1^{-}}f(x)=+\infty$
b. $\lim_{x\to1^{+}}f(x)=-\infty$
c. $\lim_{x\to1}f(x)$ does not exist
d. $\lim_{x\to2^{-}}f(x)=-\infty$
e. $\lim_{x\to2^{+}}f(x)=+\infty$
f. $\lim_{x\to2}f(x)$ does not exist