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 □
Transcript text: Which of the statements a through $k$ about the function $y=f(x)$ graphed here are true, and which are false?
a. The statement $\lim _{x \rightarrow 0} f(x)$ exists is true.
b. The statement $\lim _{x \rightarrow 0} f(x)=-2$ is $\square$
Solution
Solution Steps
Step 1: Analyze the limit as x approaches 0
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.
Step 2: Determine if statement (a) is true or false
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.
Step 3: Determine if statement (b) is true or false
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
a. True
b. True
\\(\boxed{\text{a. True, b. True}}\\)