Question

the graph of a function f is shown above. for which of the following values of c does (lim_{x \to c} f(x) = 1)?
a) 0 only
b) 0 and 3 only
c) (-2) and 0 only
d) (-2) and 3 only
e) (-2), 0, and 3

Explanation:

Step1: Recall Limit Definition

The limit \(\lim_{x \to c} f(x) = L\) means as \(x\) approaches \(c\) (from both left and right), \(f(x)\) approaches \(L\) (here \(L = 1\)).

Step2: Analyze \(c = -2\)

For \(c = -2\), as \(x\) approaches \(-2\) from left and right, the graph approaches \(y = 1\) (since the left - hand and right - hand limits at \(x=-2\) are both \(1\)).

Step3: Analyze \(c = 0\)

For \(c = 0\), the function has a peak at \(x = 0\), but the limit as \(x\to0\) is the value the function approaches around \(0\). The graph near \(x = 0\) is a semicircle, and the limit as \(x\to0\) is not \(1\) (it approaches a value greater than \(1\), the \(y\) - intercept seems to be greater than \(1\)).

Step4: Analyze \(c = 3\)

For \(c = 3\), as \(x\) approaches \(3\) from left and right, the graph (the linear part on the right) approaches \(y = 1\) (the left - hand limit as \(x\to3^{-}\) and right - hand limit as \(x\to3^{+}\) are both \(1\)).

So the values of \(c\) for which \(\lim_{x\to c}f(x)=1\) are \(c=-2\) and \(c = 3\) only.

Answer:

D. \(-2\) and \(3\) only