Question

use the graph to answer the question. describe the continuity of the graphed function. (1 point) the function has a removable discontinuity at x = 3. the function has a removable discontinuity at x = 0. the function has a nonremovable discontinuity at x = 3. the function has a nonremovable discontinuity at x = 0.

Explanation:

Step1: Recall the definition of continuity and discontinuity

A function \(y = f(x)\) is continuous at \(x=a\) if \(\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=f(a)\). A removable discontinuity occurs when \(\lim_{x
ightarrow a}f(x)\) exists but \(f(a)\) is not defined or \(f(a)
eq\lim_{x
ightarrow a}f(x)\). A non - removable (infinite or jump) discontinuity occurs when \(\lim_{x
ightarrow a^{-}}f(x)
eq\lim_{x
ightarrow a^{+}}f(x)\) (jump) or \(\lim_{x
ightarrow a^{-}}f(x)=\pm\infty\) or \(\lim_{x
ightarrow a^{+}}f(x)=\pm\infty\) (infinite).

Step2: Analyze the graph at \(x = 0\)

As \(x
ightarrow0^{-}\), \(y
ightarrow-\infty\) and as \(x
ightarrow0^{+}\), \(y
ightarrow+\infty\). So \(\lim_{x
ightarrow0^{-}}f(x)\) and \(\lim_{x
ightarrow0^{+}}f(x)\) do not exist in the finite sense (infinite discontinuity).

Step3: Analyze the graph at \(x = 3\)

The function is well - defined and smooth near \(x = 3\). There is no break, hole, or jump in the graph near \(x=3\).

Answer:

The function has a nonremovable discontinuity at \(x = 0\).