Question

answer attempt 1 out of 2 dne undefined since lim_{x→0^ - }f(x)= and lim_{x→0^ + }f(x)=, it can be determined that lim_{x→0}f(x) exists and is equal to. furthermore, it can be seen that f(0)=. therefore, it can be stated that the function f(x) is at x = 0 because

Explanation:

Step1: Find left - hand limit

As \(x\to0^{-}\), by observing the graph, we approach the value of the function from the left - hand side of \(x = 0\). The value of \(y\) approaches \(2\). So, \(\lim_{x\to0^{-}}f(x)=2\).

Step2: Find right - hand limit

As \(x\to0^{+}\), by observing the graph, we approach the value of the function from the right - hand side of \(x = 0\). The value of \(y\) approaches \(2\). So, \(\lim_{x\to0^{+}}f(x)=2\).

Step3: Determine the limit

Since \(\lim_{x\to0^{-}}f(x)=\lim_{x\to0^{+}}f(x) = 2\), then \(\lim_{x\to0}f(x)=2\).

Step4: Find the function value at \(x = 0\)

From the graph, the point on the function at \(x = 0\) has a \(y\) - value of \(2\). So, \(f(0)=2\).

Step5: Determine continuity

A function \(y = f(x)\) is continuous at \(x=a\) if \(\lim_{x\to a}f(x)=f(a)\). Here, since \(\lim_{x\to0}f(x)=2\) and \(f(0)=2\), the function \(f(x)\) is continuous at \(x = 0\).

Answer:

\(\lim_{x\to0^{-}}f(x)=2\), \(\lim_{x\to0^{+}}f(x)=2\), \(\lim_{x\to0}f(x)=2\), \(f(0)=2\), continuous