Question
Determine whether the following statement is true or false: "If \( \lim _{x \rightarrow 3} f(x)=7 \), then \( \lim _{x \rightarrow 3^{-}} f(x)=7 \).
Ask by Willis Bowen. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The statement is true.
Solution
**True.**
**Explanation:**
For the two-sided limit \(\lim_{x \rightarrow 3} f(x) = 7\) to exist, both the left-hand limit \(\lim_{x \rightarrow 3^-} f(x)\) and the right-hand limit \(\lim_{x \rightarrow 3^+} f(x)\) must exist and be equal to each other. Specifically:
1. **Two-Sided Limit Exists:** \(\lim_{x \rightarrow 3} f(x) = 7\) implies that as \(x\) approaches 3 from both the left and right, the function \(f(x)\) approaches 7.
2. **Left-Hand Limit:** Since the two-sided limit exists and equals 7, it necessarily follows that \(\lim_{x \rightarrow 3^-} f(x) = 7\).
3. **Right-Hand Limit:** Similarly, \(\lim_{x \rightarrow 3^+} f(x) = 7\).
Therefore, the statement "If \( \lim _{x \rightarrow 3} f(x)=7 \), then \( \lim _{x \rightarrow 3^{-}} f(x)=7 \)" is **true**.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
The statement is true. If the limit of \( f(x) \) as \( x \) approaches 3 is 7, it implies that both the left-hand limit, \( \lim _{x \rightarrow 3^{-}} f(x) \), and the right-hand limit, \( \lim _{x \rightarrow 3^{+}} f(x) \), must also equal 7 for the overall limit to exist. Hence, it follows that \( \lim _{x \rightarrow 3^{-}} f(x) = 7 \).
