Question

find all values x = a where the function is discontinuous. for each point of discontinuity, give (a) lim x→a - f(x), (b) lim x→a + f(x), (c) lim x→a f(x), (d) f(a) if it exists, and (e) identify which conditions for continuity are not met. be sure to note when the limit doesnt exist. f(x) is discontinuous at x = (use a comma to separate answers as needed.)

Explanation:

Step1: Recall continuity conditions

A function $f(x)$ is continuous at $x = a$ if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=\lim_{x
ightarrow a}f(x)=f(a)$. Analyze the graph for break - points.

Step2: Identify points of discontinuity

Look for jumps, holes, or vertical asymptotes on the graph. At a jump discontinuity, $\lim_{x
ightarrow a^{-}}f(x)
eq\lim_{x
ightarrow a^{+}}f(x)$. At a removable discontinuity (hole), $\lim_{x
ightarrow a}f(x)$ exists but $f(a)$ is not defined or not equal to the limit.

Answer:

(Without seeing the actual graph values, we can't give specific $x$ - values. But the general process is to find where the left - hand limit, right - hand limit, and function value don't match. If there are points $x_1,x_2,\cdots$ where the function is discontinuous, we would list them separated by commas as required by the problem.)