Evaluate [tex]\lim _{x \rightarrow 0} \frac{e^x-e^{-x}}{\sin x}=\square[/tex]
Answer (2)
To find lim x → 0 s i n x e x − e − x , we recognize it as an indeterminate form 0 0 . Using L'Hôpital's Rule, we take the derivatives of the numerator and denominator and evaluate the limit, resulting in a final value of 2. Thus, the limit is 2 .
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Recognize the limit is in indeterminate form 0 0 .
Apply L'Hôpital's Rule by differentiating the numerator and denominator.
Find the derivatives: ( e x − e − x ) ′ = e x + e − x and ( sin x ) ′ = cos x .
Evaluate the limit of the derivatives: lim x → 0 c o s x e x + e − x = 2 .
2
Explanation
Problem Analysis and Indeterminate Form We are asked to evaluate the limit of the function s i n x e x − e − x as x approaches 0 . First, we need to check if we can directly substitute x = 0 into the function. If we do so, we get s i n 0 e 0 − e − 0 = 0 1 − 1 = 0 0 , which is an indeterminate form. This means we can use L'Hôpital's Rule.
Applying L'Hôpital's Rule L'Hôpital's Rule states that if we have a limit of the form 0 0 or ∞ ∞ , we can take the derivative of the numerator and the derivative of the denominator and then evaluate the limit again. So, we need to find the derivatives of the numerator and the denominator.
Finding Derivatives The numerator is e x − e − x . Its derivative with respect to x is d x d ( e x − e − x ) = e x − ( − 1 ) e − x = e x + e − x . The denominator is sin x . Its derivative with respect to x is d x d ( sin x ) = cos x .
Evaluating the Limit Now we need to evaluate the limit of the ratio of the derivatives: lim x → 0 c o s x e x + e − x . Substituting x = 0 into this expression, we get c o s 0 e 0 + e − 0 = 1 1 + 1 = 1 2 = 2 .
Final Answer Therefore, the limit of the given function as x approaches 0 is 2.
Examples
Imagine you are analyzing the behavior of a damped oscillator where the displacement is given by a function similar to the one in this problem. Evaluating the limit as time approaches zero helps you understand the initial state of the oscillator. L'Hôpital's Rule, as applied here, is a powerful tool in physics and engineering for analyzing such systems and determining their behavior at critical points.
