MTH 101 Quiz - Limits
Evaluate the limit:
lim (x → 2) (x² - 4)
To solve this limit, we directly substitute x = 2 into the function:
2² - 4 = 4 - 4 = 0
Thus, the limit is 0.
Find the limit:
lim (x → 0) (sin x / x)
This is a well-known trigonometric limit that approaches 1 as x approaches 0. It can be
derived using the Squeeze Theorem.
Determine the limit:
lim (x → ∞) (1/x)
As x grows indefinitely, the fraction 1/x gets closer and closer to 0. Therefore, the limit is 0.
Evaluate the limit:
lim (x → 3) [(x² - 9) / (x - 3)]
First, factor the numerator:
(x - 3)(x + 3) / (x - 3)
Cancel (x - 3), leaving x + 3. Now, substituting x = 3, we get 6.
Compute the limit:
lim (x → -1) [(x³ + 1) / (x + 1)]
Factor x³ + 1 using the sum of cubes formula:
(x + 1)(x² - x + 1) / (x + 1)
Cancel (x + 1), leaving x² - x + 1. Substituting x = -1 results in 3.
Find the limit:
lim (x → ∞) [(2x² + 3) / (x² - 5)]
Divide the numerator and denominator by x²:
(2 + 3/x²) / (1 - 5/x²)
As x → ∞, the fractions vanish, leaving 2.
Evaluate the limit:
lim (x → 0) [(e^x - 1) / x]
Evaluate the limit:
lim (x → 2) (x² - 4)
To solve this limit, we directly substitute x = 2 into the function:
2² - 4 = 4 - 4 = 0
Thus, the limit is 0.
Find the limit:
lim (x → 0) (sin x / x)
This is a well-known trigonometric limit that approaches 1 as x approaches 0. It can be
derived using the Squeeze Theorem.
Determine the limit:
lim (x → ∞) (1/x)
As x grows indefinitely, the fraction 1/x gets closer and closer to 0. Therefore, the limit is 0.
Evaluate the limit:
lim (x → 3) [(x² - 9) / (x - 3)]
First, factor the numerator:
(x - 3)(x + 3) / (x - 3)
Cancel (x - 3), leaving x + 3. Now, substituting x = 3, we get 6.
Compute the limit:
lim (x → -1) [(x³ + 1) / (x + 1)]
Factor x³ + 1 using the sum of cubes formula:
(x + 1)(x² - x + 1) / (x + 1)
Cancel (x + 1), leaving x² - x + 1. Substituting x = -1 results in 3.
Find the limit:
lim (x → ∞) [(2x² + 3) / (x² - 5)]
Divide the numerator and denominator by x²:
(2 + 3/x²) / (1 - 5/x²)
As x → ∞, the fractions vanish, leaving 2.
Evaluate the limit:
lim (x → 0) [(e^x - 1) / x]
