AP Calculus AB Practice Quiz: Determining Limits Using Algebraic Manipulation

Correct Answer: B

Direct substitution of x=3 results in the indeterminate form 0/0. To find the limit, we must first create an equivalent expression by factoring the numerator. The expression x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3). The (x - 3) term in the numerator cancels with the term in the denominator, leaving the equivalent expression x + 3. Now, we can evaluate the limit by substituting x = 3 into the simplified expression: 3 + 3 = 6.

Correct Answer: A

Direct substitution yields the indeterminate form 0/0. To evaluate this limit, we can use the algebraic technique of multiplying the numerator and denominator by the conjugate of the numerator, which is sqrt(x) + 2. This gives: lim [(sqrt(x) - 2)(sqrt(x) + 2)] / [(x - 4)(sqrt(x) + 2)]. The numerator simplifies to (x - 4). The (x - 4) terms cancel, leaving the equivalent expression 1 / (sqrt(x) + 2). Substituting x = 4 gives 1 / (sqrt(4) + 2) = 1 / (2 + 2) = 1/4.

Correct Answer: A

This limit results in the indeterminate form 0/0 upon direct substitution. To find the limit, we need to rearrange the complex fraction into an equivalent form. Find a common denominator for the terms in the numerator, which is 5(x+5). This gives: lim ( (5 - (x+5)) / (5(x+5)) ) / x. The numerator simplifies to -x. The expression becomes lim (-x / (5(x+5))) / x. The x in the numerator cancels with the x in the main denominator, leaving the equivalent expression -1 / (5(x+5)). Now, substitute x = 0: -1 / (5(0+5)) = -1/25.

Correct Answer: B

This problem uses the Squeeze Theorem. We evaluate the limits of the bounding functions as x approaches 1. For the lower bound: lim (2x^2 + 3) as x->1 is 2(1)^2 + 3 = 5. For the upper bound: lim (x^4 - 2x^2 + 6) as x->1 is (1)^4 - 2(1)^2 + 6 = 1 - 2 + 6 = 5. Since g(x) is squeezed between two functions that both approach 5 as x approaches 1, the Squeeze Theorem states that the limit of g(x) as x approaches 1 must also be 5.

Correct Answer: B

Direct substitution of x=2 results in the indeterminate form 0/0. We must create an equivalent expression by factoring both the numerator and the denominator. The numerator is a difference of cubes, x^3 - 2^3, which factors to (x - 2)(x^2 + 2x + 4). The denominator is a quadratic that factors to (x - 2)(x + 1). The common factor (x - 2) cancels out. The equivalent expression is (x^2 + 2x + 4) / (x + 1). Evaluating the limit by substituting x = 2 into this simplified expression gives (2^2 + 2(2) + 4) / (2 + 1) = (4 + 4 + 4) / 3 = 12 / 3 = 4.

Correct Answer: A

Substituting x=2 gives 0/0, an indeterminate form. To find an equivalent expression, we can factor the numerator. 4 - x^2 can be factored as -(x^2 - 4), which is -(x - 2)(x + 2). The expression becomes lim (-(x - 2)(x + 2)) / (x - 2). The (x - 2) terms cancel, leaving the equivalent expression -(x + 2). Substituting x = 2 into this expression gives -(2 + 2) = -4.

Correct Answer: C

Direct substitution results in the indeterminate form 0/0. We can find an equivalent expression by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(x+3) + 2. This gives: lim [(x - 1)(sqrt(x+3) + 2)] / [(sqrt(x+3) - 2)(sqrt(x+3) + 2)]. The denominator simplifies to (x+3) - 4, which is (x - 1). The (x - 1) terms in the numerator and denominator cancel out, leaving the equivalent expression sqrt(x+3) + 2. Now, we can substitute x = 1: sqrt(1+3) + 2 = sqrt(4) + 2 = 2 + 2 = 4.