Question

the graph of ( y = f(x) ) is shown above (graph of f). what is ( lim_{x \to 0} f(x) )?
options:
a) 0
b) 1
c) 3
d) the limit does not exist.

Explanation:

Step1: Recall limit definition

The limit of a function \( \lim_{x \to a} f(x) \) exists if the left - hand limit and the right - hand limit as \( x \) approaches \( a \) are equal. For \( \lim_{x \to 0} f(x) \), we need to check the behavior of \( f(x) \) as \( x \) approaches 0 from both the left (\( x\to0^{-} \)) and the right (\( x\to0^{+} \)).

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

From the given graph of \( y = f(x) \), when we approach \( x = 0 \) from the left (values of \( x \) slightly less than 0) and from the right (values of \( x \) slightly greater than 0), the \( y \) - value that the graph approaches is 1. Since the left - hand limit and the right - hand limit as \( x \to 0 \) are both equal to 1, by the definition of the limit, \( \lim_{x \to 0} f(x)=1 \).

Answer:

B. 1