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Limits and Continuity Worksheet
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Limits and Continuity Worksheet
📚 Part 1: Evaluating Limits
1. Evaluate the following limit: lim(x→3) (x² - 9)/(x - 3)
0
6
9
Does not exist
2. Find lim(x→∞) (3x² + 2x - 1)/(x² - 5)
3. Evaluate lim(x→0) (sin(5x))/(x)
0
1
5
Does not exist
4. Calculate lim(x→2⁻) (x + 1)/(x - 2) and lim(x→2⁺) (x + 1)/(x - 2)
✏️ Part 2: Continuity Analysis
5. A function f(x) is continuous at x = a if which conditions are satisfied? (Check all that apply)
f(a) is defined
lim(x→a) f(x) exists
lim(x→a) f(x) = f(a)
f'(a) exists
6. Consider f(x) = { x² + 1 for x < 2, kx - 1 for x ≥ 2 }. Find the value of k that makes f(x) continuous at x = 2.
7. Determine where g(x) = (x² - 4)/(x + 2) is discontinuous and classify the type of discontinuity.
8. True or False: If lim(x→a) f(x) exists, then f(x) must be continuous at x = a.
Answer: _____________ Explain your reasoning:
🎯 Part 3: Application Problems
9. The temperature T(t) in degrees Celsius of a cooling object is given by T(t) = 20 + 60e^(-0.1t), where t is time in minutes. Find lim(t→∞) T(t) and interpret its meaning.
10. A particle's position is given by s(t) = { t² for 0 ≤ t < 3, 18 - 3t for t ≥ 3 }. Is the position function continuous at t = 3? Show all work.
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