Question

Consider the function: \(f(x)=\begin{cases} x^{2}+1 & \text{if }x\lt 2 \\ 5 & \text{if }x=2 \\ 3x-1 & \text{if }x>2 \end{cases}\)
Which of the following statements is correct ?

• A. \(\lim _{x\rightarrow 2}f(x)\) exists and is equal to \(5\)
• B. \(\lim _{x\rightarrow 2}f(x)\) exists but is not equal to \(5\)
• C. \(\lim _{x\rightarrow 2}f(x)\) does not exist because \(f(2)\) is defined
• D. \(\lim _{x\rightarrow 2}f(x)\) does not exist because left-hand and right-hand limits are not equal

Hint:

Compare the left and right limits at 2; the value of f at 2 does not affect the limit.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A limit exists at a point if the left-hand and right-hand limits are equal; the value of the function at the point is not involved.

Step 2: Left-hand limit
\(\lim_{x\to 2^{-}}(x^{2}+1)=4+1=5\)

Step 3: Right-hand limit
\(\lim_{x\to 2^{+}}(3x-1)=6-1=5\)

Step 4: Conclusion
Both are 5, so \(\lim_{x\to 2}f(x)=5\), which also equals \(f(2)=5\).

Final Answer:
The limit exists and equals 5. \[ \boxed{\text{(A) }\lim _{x\rightarrow 2}f(x)\text{ exists and is equal to }5} \]