▶️ Answer/Explanation
Solution
1. Left-hand limit (approaching 2 from below):
As \( x \) approaches 2 from the left (\( x \to 2^- \)), \( f(x) \) approaches approximately 8.
Values: 7.80, 7.86, 7.90, 7.95
2. Right-hand limit (approaching 2 from above):
As \( x \) approaches 2 from the right (\( x \to 2^+ \)), \( f(x) \) approaches approximately 8.
Values: 8.05, 8.10, 8.14, 8.20
3. Function value at x=2:
\( f(2) = 3 \) (which is different from the limit values)
4. Conclusion:
– The left and right limits both approach 8, suggesting \(\lim_{x \to 2} f(x) = 8\)
– However, the function value at x=2 is different (3)
– The table doesn’t show values arbitrarily close to 2, so we cannot be certain about the limit
– Therefore, the limit cannot be definitively determined from the given data
✅ Answer: D) \(\lim_{x \to 2} f(x)\) cannot be definitively determined from the data in the table.
▶️ Answer/Explanation
Solution
1. Left-hand limit (approaching 5 from below):
As \( x \) approaches 5 from the left (\( x \to 5^- \)), \( f(x) \) becomes increasingly negative:
Values: 4, -16, -256, -726 (no approach to 8)
2. Right-hand limit (approaching 5 from above):
As \( x \) approaches 5 from the right (\( x \to 5^+ \)), \( f(x) \) approaches 8:
Values: 7.9999, 7.999, 7.99, 7.9 (clearly approaching 8)
3. Two-sided limit:
Since the left-hand and right-hand limits are different, \(\lim_{x \to 5} f(x)\) does not exist
4. Option analysis:
– A) False (two-sided limit doesn’t exist)
– B) False (left-hand limit doesn’t approach 8)
– C) True (right-hand limit clearly approaches 8)
– D) False (irrelevant to the data shown)
✅ Answer: C) \(\lim_{x \to 5^+} f(x)=8\)
▶️ Answer/Explanation
Solution
1. Left-hand limit (approaching 4 from below):
As \( x \) approaches 4 from the left (\( x \to 4^- \)), \( f(x) \) approaches approximately 7:
Values: 7.018, 7.007, 7.002 (clearly approaching 7)
2. Right-hand limit (approaching 4 from above):
For values very close to 4 from the right (4.001), \( f(x) = 6.998 \) (still close to 7)
3. Best approximation:
– The immediate values closest to 4 (3.999 and 4.001) show \( f(x) \) approaching 7
– The function is stated to be continuous
4. Conclusion:
The best approximation for \(\lim_{x \to 4} f(x)\) is 7
✅ Answer: C) 7
▶️ Answer/Explanation
Solution
1. Direct Substitution:
\[ \frac{(-3)+3}{(-3)^2+9} = \frac{0}{9+9} = \frac{0}{18} = 0 \]
Since we don’t get an indeterminate form (like 0/0), direct substitution works.
2. Analysis:
– Numerator approaches 0 as t → -3
– Denominator approaches 18 as t → -3 (never zero)
– Therefore, the limit is 0/18 = 0
✅ Answer: B) 0