▶️ Answer/Explanation
Solution
Correct Answer: C
As \( x \to -\infty \):
\( 2^x \to 0 \)
\( 5^x \to 0 \)
Limit becomes \( \frac{3}{4} \)
Why other options are wrong:
A) -2/5: Incorrect value
B) 0: Limit is not zero
D) nonexistent: Limit is finite
\( 2^x \to 0 \)
\( 5^x \to 0 \)
Limit becomes \( \frac{3}{4} \)
Why other options are wrong:
A) -2/5: Incorrect value
B) 0: Limit is not zero
D) nonexistent: Limit is finite
▶️ Answer/Explanation
Solution
Correct Answer: D
The limit \( \lim_{x \to 1} f(x) \) is stated as nonexistent.
Note: The graph shows the left-hand limit (\( x \to 1^- \)) and right-hand limit (\( x \to 1^+ \)) both equal 1, suggesting the limit is 1. However, per the given answer, the limit does not exist.
Why other options are wrong:
A) 0: Limit does not equal 0
B) 1: Per given answer, limit does not exist
C) 3: Limit does not equal 3
Note: The graph shows the left-hand limit (\( x \to 1^- \)) and right-hand limit (\( x \to 1^+ \)) both equal 1, suggesting the limit is 1. However, per the given answer, the limit does not exist.
Why other options are wrong:
A) 0: Limit does not equal 0
B) 1: Per given answer, limit does not exist
C) 3: Limit does not equal 3
▶️ Answer/Explanation
Solution
Correct Answer: C
Analysis of each option:
Option A: limx→1 f(x) = 0
✓ True. The graph shows f(x) approaches 0 as x approaches 1.
Option B: limx→2 g(x) does not exist
✓ True. Left limit (1) ≠ right limit (0) at x=2.
Option C: limx→1 (f(x)g(x+1)) does not exist
✕ False. While g(x+1) has no limit at x=1, f(x)→0 makes the product→0.
Option D: limx→1 (f(x+1)g(x)) exists
✓ True. Both f(x+1)→1 and g(x)→1 as x→1, so product→1.
Conclusion: Option C is the false statement.