Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (tan 5x) / x

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Theorem 3.10 refers to the limit of the form lim x→0 (sin(ax)/x) = a, which is a standard result in calculus.

To use this theorem, we need to express tan(5x) in terms of sine and cosine: tan(5x) = sin(5x)/cos(5x).

Rewrite the original limit as lim x→0 (sin(5x)/cos(5x)) / x, which can be rearranged to lim x→0 (sin(5x)/(x * cos(5x))).

Separate the limit into two parts: lim x→0 (sin(5x)/x) * lim x→0 (1/cos(5x)).

Apply Theorem 3.10 to the first part: lim x→0 (sin(5x)/x) = 5, and evaluate the second part: lim x→0 (1/cos(5x)) = 1, since cos(0) = 1.

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주요 개념

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 0 helps determine the behavior of the function tan(5x)/x near that point.

Theorem 3.10 (Limit of a Trigonometric Function)

Theorem 3.10 typically refers to a specific limit involving trigonometric functions, often stating that lim x→0 (sin x)/x = 1. This theorem can be extended to other functions, such as tan(5x), by recognizing that tan(x) behaves similarly to sin(x) near zero, allowing us to simplify the limit evaluation.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that if such a form occurs, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can be applied to the limit in the question if direct substitution leads to an indeterminate form.

Use Theorem 3.10 to evaluate the following limits.lim x🠂0 (tan 5x) / x