Questions: Which of the statements a through k about the function y=f(x) graphed here are true, and which are false? a. The statement lim x → 0 f(x) exists is true. b. The statement lim x → 0 f(x)=-2 is □

Solution Steps

As x approaches 0 from the left, the function f(x) approaches -2. As x approaches 0 from the right, the function f(x) approaches -2. Since the left-hand limit and the right-hand limit are equal, the limit exists.

The statement $\lim_{x \rightarrow 0} f(x)$ exists is true because both the left-hand and right-hand limits as x approaches 0 are equal to -2.

Since $\lim_{x \rightarrow 0^-} f(x) = -2$ and $\lim_{x \rightarrow 0^+} f(x) = -2$, then $\lim_{x \rightarrow 0} f(x) = -2$. Thus, the statement $\lim_{x \rightarrow 0} f(x) = -2$ is true.

Final Answer