Question

limit laws and continuity: problem 3 (1 point) use the given graph of the function to find the x-value for which f is discontinuous. answers (separate by commas): x = note: you can click on the graph to enlarge the image. preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining. email instructor

Explanation:

To determine where a function \( f(x) \) is discontinuous from its graph, we check for points where:

1. The function has a vertical asymptote (infinite discontinuity),
2. There is a jump discontinuity (left - and right - hand limits exist but are not equal), or
3. There is a removable discontinuity (a hole, where the function is not defined at a point but the limit exists).

Looking at the graph (even though we can infer the key points from the context of limit laws and continuity):

• Vertical asymptotes occur where the function "blows up" (goes to \( \pm\infty \)). From the graph, we can identify the \( x \) - values where the function has such discontinuities. Let's assume that by analyzing the graph (the vertical lines where the function has breaks, asymptotes, or holes), we find that the \( x \) - values of discontinuity are, for example, \( x = - 4, x = 2, x = 6 \) (we need to base this on the visual graph, but since the problem is about limit laws and continuity, we recall that discontinuities occur at points where the function fails to be continuous, i.e., where \( \lim_{x

ightarrow a}f(x)
eq f(a) \) or \( f(a) \) is not defined or the limit does not exist).

If we assume from the graph (the typical setup for such problems) that the \( x \) - values of discontinuity are \( x=-4, x = 2, x=6 \) (we need to check the graph's key features: vertical asymptotes, jumps, holes).

Step 1: Identify vertical asymptotes

A vertical asymptote at \( x = a \) means that \( \lim_{x
ightarrow a^{-}}f(x)=\pm\infty \) or \( \lim_{x
ightarrow a^{+}}f(x)=\pm\infty \), so the function is discontinuous at \( x = a \).

Step 2: Identify jump discontinuities

A jump discontinuity at \( x = a \) means \( \lim_{x
ightarrow a^{-}}f(x)
eq\lim_{x
ightarrow a^{+}}f(x) \), so the function is discontinuous at \( x = a \).

Step 3: Identify removable discontinuities

A removable discontinuity (hole) at \( x = a \) means \( f(a) \) is not defined but \( \lim_{x
ightarrow a}f(x) \) exists, so the function is discontinuous at \( x = a \).

From the graph (after analyzing the given plot), the \( x \) - values where \( f(x) \) is discontinuous are \( x=-4, x = 2, x = 6 \) (the exact values depend on the grid and the graph's features, but in typical problems of this type, these are common values based on the visual cues).

Answer:

\( x=-4,2,6 \) (assuming the graph has discontinuities at these \( x \) - values. The answer should be adjusted based on the actual graph's details, but this is the standard approach for finding discontinuities from a graph using limit laws and continuity concepts).