Question

sketch a possible graph of a function that satisfies the following conditions at x = 1 and discuss the continuity of f at x = 1. f(1)=2 and lim_{x→1} f(x)= - 2. choose the correct graph below. is the function continuous at x = 1? a. the function is discontinuous at x = 1 since lim_{x→1} f(x)≠f(1). b. the function is continuous. c. the function is discontinuous at x = 1 since f(1) does not exist. d. the function is discontinuous at x = 1 since lim_{x→1} f(x) does not exist.

Explanation:

Step1: Recall continuity condition

A function $f(x)$ is continuous at $x = a$ if $\lim_{x
ightarrow a}f(x)=f(a)$. Here $a = 1$, $f(1)=2$ and $\lim_{x
ightarrow 1}f(x)= - 2$. Since $\lim_{x
ightarrow 1}f(x)
eq f(1)$, the function is discontinuous at $x = 1$.

Step2: Analyze graph - function value

The point $(1,2)$ should be on the graph (because $f(1) = 2$).

Step3: Analyze graph - limit value

As $x$ approaches $1$, the function values should approach $- 2$.

Answer:

The correct answer for the graph is the one where the point $(1,2)$ is on the graph and the function approaches $y=-2$ as $x$ approaches $1$. For the continuity question: A. The function is discontinuous at $x = 1$ since $\lim_{x
ightarrow 1}f(x)
eq f(1)$.