Question

Which of the following statements about the function $y=f\left(x\right)$ graphed here are true,

• A. ${lim}_{x\to 2}f\left(x\right)$ does not exist
• B. ${lim}_{x\to 2}f\left(x\right)=1$
• C. ${lim}_{x\to 1}f\left(x\right)$ does not exist
• D. ${lim}_{x\to x_0}f\left(x\right)$ exists at every point $x_0$ in $(-1,1)$

Answer 1

The correct option is A ${lim}_{x\to 2}f\left(x\right)$ does not exist

(A) is true as

$\underset{x\to 2^{+}}{lt}\ne \underset{x\to 2^{-}}{lt}\ne f\left(2\right)$

(B) $\underset{x\to 2}{lt}f\left(x\right)=1$

(incorrect from graph)

(C) Correct $\underset{x\to 1}{lt}f\left(x\right)$ doesn't exist because

${lim}_{x\to 1^{-}}\ne {lim}_{x\to 1^{+}}$ and can be seen from graph

(D) is not true because ${lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0^{-}}\ne f\left(0\right)$