Selecting Procedures for Determining Limits
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AP Calculus BC › Selecting Procedures for Determining Limits
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\tan x- x}{x^3}$?
Use the method of undetermined coefficients
Use the Fundamental Theorem of Calculus Part 2
Use the Intermediate Value Theorem
Use completing the square
Use a Maclaurin series expansion
Explanation
Selecting the appropriate method for evaluating limits is a key skill in calculus. Using a Maclaurin series expansion is most efficient because expanding tan x as x + (1/3)x³ + higher terms subtracts x to give (1/3)x³ / x³ = 1/3 in the limit. This handles the 0/0 form by canceling lower-order terms. Series are powerful for higher-order indeterminate forms. A tempting distractor might be using the Intermediate Value Theorem, but that proves existence of roots, not limit values. When limits involve trigonometric functions requiring higher precision, opt for series expansions to reveal the behavior.
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\sin(2x)}{\sin(5x)}$?
Use completing the square
Use standard small-angle limits to rewrite the ratio
Apply the Mean Value Theorem for integrals
Use integration by substitution
Apply the Rational Root Theorem
Explanation
Selecting the appropriate method for evaluating limits is a key skill in calculus. Using standard small-angle limits to rewrite the ratio is most efficient because it's (sin(2x)/(2x)) / (sin(5x)/(5x)) * (2/5), and each part approaches 1, yielding 2/5. This leverages known limits directly. It's quick for ratios of sines. A tempting distractor might be applying the Mean Value Theorem for integrals, but that's for average values, not limits. When limits involve ratios of sine functions, rewrite using the sin(theta)/theta standard limit.
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\sqrt{1+x}-\sqrt{1-x}}{x}$?
Use separation of variables
Use the quadratic formula
Multiply by a conjugate to eliminate radicals
Use the Divergence Test
Compute a definite integral
Explanation
Selecting the appropriate procedure for determining limits is a key skill in AP Calculus BC, as it involves recognizing the form of the limit and choosing the most efficient method. Multiplying by a conjugate to eliminate radicals is most appropriate, as (√(1+x) - √(1-x)) / x becomes [(1+x) - (1-x)] / [x (√(1+x) + √(1-x))] = 2 / (√(1+x) + √(1-x)) → 1 as x → 0. This rationalizes the numerator efficiently. It resolves the 0/0 form algebraically. A tempting distractor might be using separation of variables, but that fails for a limit, not a DE. When limits involve differences of square roots, conjugating is a key strategy to simplify.
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\sin x}{x\sqrt{1+x}}$?
Use the Law of Sines
Use the quadratic formula
Apply the Integral Test
Use partial fractions
Separate factors and use known limits for $\sin x/x$ and $\sqrt{1+x}$
Explanation
Selecting the appropriate procedure for determining limits is a key skill in AP Calculus BC, as it involves recognizing the form of the limit and choosing the most efficient method. Separating factors and using known limits for sin x/x and √(1+x) is most appropriate: (sin x / x) / √(1+x) → 1 / 1 =1. This decomposition is efficient. Direct application. A tempting distractor might be partial fractions, but fails. Factor limits into known parts when possible.
Which method is most appropriate for evaluating $\lim_{x\to \infty}\dfrac{3x^2-5}{2x^2+7x}$?
Apply the Chain Rule
Use the quadratic formula
Compute a definite integral
Divide numerator and denominator by the highest power of $x$
Use a trig substitution
Explanation
Selecting the appropriate method for evaluating limits is a key skill in calculus. Dividing numerator and denominator by the highest power of x is most efficient because for x→∞, dividing by x² gives (3 - 5/x²)/(2 + 7/x), approaching 3/2. This reveals the horizontal asymptote quickly. It's standard for rational functions at infinity. A tempting distractor might be applying the Chain Rule, but that's for derivatives, not limits at infinity. For limits at infinity of rational functions, always divide by the highest power to simplify.
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\ln(1+x)-x}{x^2}$?
Use partial fractions
Compute a definite integral by substitution
Apply the Divergence Test
Use the Law of Sines
Use a Maclaurin series or repeated L'Hôpital's Rule
Explanation
Selecting the appropriate procedure for determining limits is a key skill in AP Calculus BC, as it involves recognizing the form of the limit and choosing the most efficient method. Using a Maclaurin series or repeated L'Hôpital's Rule is most appropriate for [ln(1+x) - x]/x², series ln(1+x)=x - x²/2 + x³/3 -..., so (x - x²/2 + ... - x)/x² = -1/2 + x/3 → -1/2; or L'Hôpital twice: 0/0, (1/(1+x) -1)/(2x) still 0/0, (-1/(1+x)²)/2 → -1/2. This handles higher-order indeterminate form efficiently. It is necessary when basic methods fail. A tempting distractor might be partial fractions, but that fails for non-rational functions. For limits with logs and powers, series or repeated L'Hôpital offers reliable strategies.
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\sin^2 x}{x^2}$?
Apply the Divergence Test
Rewrite using $\left(\dfrac{\sin x}{x}\right)^2$
Use the Shell Method
Use the quadratic formula
Use partial fractions
Explanation
Selecting the appropriate method for evaluating limits is a key skill in calculus. Rewriting using (sin x / x)² is most efficient because sin²x / x² = (sin x / x)² →1²=1, leveraging the standard limit. It's a direct application for squared terms. This simplifies without further computation. A tempting distractor might be using partial fractions, but that's for rational decomposition, not trig. For powers of sine over powers of x, rewrite as powers of the standard limit.
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\ln(1+x)}{x}$?
Apply the Pythagorean theorem
Use a midpoint Riemann sum
Use a t-distribution table
Use the Shell Method
Recognize a derivative definition at $x=0$
Explanation
Selecting the appropriate method for evaluating limits is a key skill in calculus. Recognizing a derivative definition at x=0 is most efficient because this is f'(0) for f(x)=ln(1+x), which is 1/(1+0)=1. It connects directly to differentiation. This method is precise and conceptual. A tempting distractor might be using a midpoint Riemann sum, but that's for approximations of integrals, not limits. For limits matching derivative forms, interpret them as derivatives for efficient computation.
Which method is most appropriate for evaluating $\lim_{x\to -2}\dfrac{x^2+4x+4}{x+2}$?
Factor the numerator as a perfect square and cancel
Use a tangent-line approximation at $x=1$
Use polar coordinates
Apply the Root Test
Use integration by parts
Explanation
Selecting the appropriate method for evaluating limits is a key skill in calculus. Factoring the numerator as a perfect square and canceling is most efficient because x²+4x+4=(x+2)², so (x+2)²/(x+2)=(x+2)→0 at x=-2. It simplifies the 0/0 form algebraically. This is direct for perfect squares. A tempting distractor might be using a tangent-line approximation at x=1, but that's for linear approximations, not exact limits. When numerators are perfect squares, factor and cancel before other methods.
Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\sin x - x}{x^3}$?
Use the Law of Cosines
Use a slope field
Apply the Divergence Test
Use substitution in a definite integral
Use a Maclaurin series expansion
Explanation
Selecting the appropriate method for evaluating limits is a key skill in calculus. Using a Maclaurin series expansion is most efficient because sin x = x - x³/6 + higher, so (x - x³/6 - x)/x³ = -1/6. It cancels lower terms to reveal the limit. Series handle higher-order zeros well. A tempting distractor might be using substitution in a definite integral, but that's for integration, not limits. For trig limits requiring precision beyond first order, employ Maclaurin series expansions.