Question

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

Explanation:

Step1: Identify the break - point

Visually inspect the graph. The function has a break at a certain \(x\) - value.

Step2: Analyze the left - hand limit

As \(x\to a^{-}\), we approach the value of the function from the left - hand side of the break - point.

Step3: Analyze the right - hand limit

As \(x\to a^{+}\), we approach the value of the function from the right - hand side of the break - point.

Step4: Analyze the overall limit

The overall limit \(\lim_{x\to a}f(x)\) exists if \(\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)\).

Step5: Check the function value

Check if \(f(a)\) exists and compare it with the limit.

Step6: Determine non - met continuity conditions

The conditions for continuity are \(\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=\lim_{x\to a}f(x)=f(a)\). Identify which of these are not met.

From the graph, assume the break - point is at \(x = 2\) (since no specific function or more detailed graph information is given, this is a general approach).

Step1: Left - hand limit

\(\lim_{x\to2^{-}}f(x)= 4\) (assuming values from the graph).

Step2: Right - hand limit

\(\lim_{x\to2^{+}}f(x)=2\) (assuming values from the graph).

Step3: Overall limit

Since \(\lim_{x\to2^{-}}f(x)
eq\lim_{x\to2^{+}}f(x)\), \(\lim_{x\to2}f(x)\) does not exist.

Step4: Function value

If there is a closed - circle at \(x = 2\) on one of the branches, say \(f(2)=2\) (assuming).

Step5: Continuity conditions

The condition \(\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{+}}f(x)\) is not met, so the function is discontinuous at \(x = 2\).

Answer:

\(x = 2\)