Use l’Hôpital’s rule to find the limits in Exercises 7–52.
27. lim (x → (π/2)^-) (x - π/2) sec x

검증된 단계별 안내

Identify the limit expression: \(\lim_{x \to (\pi/2)^-} (x - \pi/2) \sec x\).

Check the form of the limit by substituting \(x = \pi/2\) from the left side: \((x - \pi/2)\) approaches 0, and \(\sec x = \frac{1}{\cos x}\) tends to \(\pm \infty\) because \(\cos(\pi/2) = 0\). This suggests an indeterminate form of type \(0 \cdot \infty\).

Rewrite the expression to apply l’Hôpital’s rule by converting the product into a quotient. For example, write it as \(\frac{x - \pi/2}{\cos x}\) because \(\sec x = \frac{1}{\cos x}\).

Now the limit becomes \(\lim_{x \to (\pi/2)^-} \frac{x - \pi/2}{\cos x}\), which is of the form \(\frac{0}{0}\), suitable for l’Hôpital’s rule.

Apply l’Hôpital’s rule by differentiating numerator and denominator separately: differentiate numerator \(\frac{d}{dx}(x - \pi/2) = 1\), and denominator \(\frac{d}{dx}(\cos x) = -\sin x\). Then evaluate the new limit \(\lim_{x \to (\pi/2)^-} \frac{1}{-\sin x}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits and One-Sided Limits

A limit describes the value a function approaches as the input approaches a certain point. One-sided limits consider the approach from only one side, such as from the left (denoted by the minus sign). Understanding how to evaluate these limits is essential for analyzing behavior near points where the function may be undefined.

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 the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met. This rule simplifies complex limit problems.

Behavior of Trigonometric Functions Near Critical Points

Trigonometric functions like sec(x) can have vertical asymptotes or undefined points at specific values, such as π/2 for sec(x). Understanding how these functions behave near such points helps in correctly applying limit techniques and interpreting the function’s behavior as x approaches these critical values.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.27. lim (x → (π/2)^-) (x - π/2) sec x