Selecting the appropriate method for evaluating limits is a key skill in calculus. Using the Squeeze Theorem with unit-circle inequalities is most efficient because -|x| ≤ sin x ≤ |x| for small x, dividing by |x| and taking limits yields 1, proving sin x / x → 1. This bounds the function rigorously. It's foundational for trig limits. A tempting distractor might be completing the square, but that's for quadratics, not trig functions. For proving fundamental limits like sin x / x, apply the Squeeze Theorem using geometric inequalities.

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.

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.

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. Dividing numerator and denominator by x and using sin x/x →1 is most appropriate: sin x / (x + sin x) = (sin x / x) / (1 + sin x / x) → 1 / (1+1) = 1/2. This simplifies using standard limits. It is efficient near 0. A tempting distractor might be completing the square, but that fails without quadratics. For trig over linear plus trig, normalize by dividing by x to apply known limits.

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.

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 the conjugate is most appropriate here to rationalize the numerator, simplifying (√(1+x) - 1)/x to 1/(√(1+x) + 1), which evaluates to 1/2 at x=0. This algebraic manipulation efficiently resolves the 0/0 indeterminate form. It avoids unnecessary trigonometric identities or other techniques not applicable here. A tempting distractor might be using a trig identity for sin(2x), but that fails because no trigonometric functions are present. For limits with square roots causing indeterminate forms, rationalizing via the conjugate is often the most straightforward strategy.

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.

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.

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.

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 known limit for $\ln(1+u)/u$ with substitution $u = 2x$ is most appropriate because as x approaches 0, it transforms into the standard limit that equals 1, multiplied by 2. This method leverages fundamental limit properties without needing differentiation or series expansions. It is efficient for indeterminate forms involving logarithms near 1. A tempting distractor might be partial fraction decomposition, but that fails because the expression is not a rational function suitable for decomposition. When encountering limits with logarithms approaching 0, consider substituting to match known limit forms for quick resolution.