Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
71. lim (x → (π/2)⁻) sec x / tan x

검증된 단계별 안내

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

Rewrite the trigonometric functions in terms of sine and cosine to simplify the expression: \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\). Substitute these into the limit to get \(\lim_{x \to (\pi/2)^-} \frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}}\).

Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}} = \frac{1}{\cos x} \times \frac{\cos x}{\sin x} = \frac{1}{\sin x}\).

Now the limit reduces to \(\lim_{x \to (\pi/2)^-} \frac{1}{\sin x}\). Consider the behavior of \(\sin x\) as \(x\) approaches \(\pi/2\) from the left side.

Since \(\sin x\) approaches 1 as \(x\) approaches \(\pi/2\), analyze the limit \(\frac{1}{\sin x}\) accordingly to determine the limit value.

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주요 개념

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

Behavior of Trigonometric Functions Near Specific Points

Understanding how functions like sec x and tan x behave as x approaches π/2 from the left is crucial. Near π/2, tan x tends to infinity, while sec x also grows large but their rates differ, affecting the limit's value.

Limits Involving Indeterminate Forms and Alternative Techniques

When L’Hôpital’s Rule leads to cyclic or unresolvable forms, alternative methods such as algebraic manipulation, trigonometric identities, or analyzing one-sided limits must be used to evaluate the limit.

One-Sided Limits and Directional Approach

Evaluating limits as x approaches π/2 from the left (denoted π/2⁻) requires considering the function's behavior just before π/2, which can differ from approaching from the right, impacting the limit's existence and value.

Theory and ApplicationsL’Hôpital’s Rule does not help with the limits in Exercises 69–76. Try it—you just keep on cycling. Find the limits some other way.71. lim (x → (π/2)⁻) sec x / tan x