Evaluate the limit using L'Hospital's rule
$\lim _{x \rightarrow 0} \frac{e^x-1}{\sin (8 x)}$
Answer (2)
By applying L'Hospital's rule, we find that the limit lim x → 0 s i n ( 8 x ) e x − 1 = 8 1 . This is determined by checking the indeterminate form and calculating the derivatives of the numerator and denominator. The final result of the limit is 8 1 .
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Check that the limit is in indeterminate form 0 0 .
Apply L'Hospital's rule by finding the derivatives of the numerator and denominator: d x d ( e x − 1 ) = e x and d x d ( sin ( 8 x )) = 8 cos ( 8 x ) .
Evaluate the limit of the ratio of the derivatives: lim x → 0 8 c o s ( 8 x ) e x = 8 c o s ( 0 ) e 0 = 8 1 .
Conclude that lim x → 0 s i n ( 8 x ) e x − 1 = 8 1 .
Explanation
Checking Indeterminate Form We are asked to evaluate the limit lim x → 0 s i n ( 8 x ) e x − 1 using L'Hospital's rule. First, we need to check if the limit is in an indeterminate form. As x approaches 0 , e x − 1 approaches e 0 − 1 = 1 − 1 = 0 , and sin ( 8 x ) approaches sin ( 0 ) = 0 . Thus, the limit is in the indeterminate form 0 0 , and we can apply L'Hospital's rule.
Applying L'Hospital's Rule L'Hospital's rule states that if lim x → c f ( x ) = 0 and lim x → c g ( x ) = 0 , and if lim x → c g ′ ( x ) f ′ ( x ) exists, then lim x → c g ( x ) f ( x ) = lim x → c g ′ ( x ) f ′ ( x ) . In our case, f ( x ) = e x − 1 and g ( x ) = sin ( 8 x ) . We need to find the derivatives of f ( x ) and g ( x ) .
Finding Derivatives The derivative of f ( x ) = e x − 1 with respect to x is f ′ ( x ) = e x . The derivative of g ( x ) = sin ( 8 x ) with respect to x is g ′ ( x ) = 8 cos ( 8 x ) .
Evaluating the Limit Now we need to evaluate the limit of the ratio of the derivatives: lim x → 0 8 c o s ( 8 x ) e x . As x approaches 0 , e x approaches e 0 = 1 , and cos ( 8 x ) approaches cos ( 0 ) = 1 . Therefore, the limit is 8 × 1 1 = 8 1 .
Final Answer Thus, by L'Hospital's rule, lim x → 0 s i n ( 8 x ) e x − 1 = 8 1 .
Examples
L'Hospital's rule is a powerful tool used in various fields like physics and engineering to evaluate limits that arise in indeterminate forms. For example, when analyzing the behavior of electrical circuits or fluid dynamics, you might encounter expressions that become indeterminate at certain points. Applying L'Hospital's rule helps in determining the true behavior of these systems near those critical points, allowing for accurate modeling and predictions. Understanding and applying L'Hospital's rule is essential for solving complex problems in these fields.
