Evaluate the limit using L'Hospital's rule
$\lim _{x \rightarrow \infty} 14 x\left(e^{\frac{1}{x}}-1\right)$
Answer (2)
The limit lim x → ∞ 14 x ( e x 1 − 1 ) can be rewritten using a substitution that allows the use of L'Hospital's rule. After applying the rule, we find that the limit evaluates to 14 . Therefore, the final answer is 14.
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Rewrite the limit as lim x → ∞ 14 x 1 e x 1 − 1 .
Substitute u = x 1 , transforming the limit to lim u → 0 14 u e u − 1 .
Apply L'Hospital's rule to get 14 lim u → 0 1 e u .
Evaluate the limit to find the final answer: 14 .
Explanation
Problem Setup We are asked to evaluate the limit lim x → ∞ 14 x ( e x 1 − 1 ) using L'Hospital's rule.
Rewriting the Expression First, we rewrite the expression to make it suitable for L'Hospital's rule. We can rewrite the expression as x → ∞ lim 14 x ( e x 1 − 1 ) = x → ∞ lim 14 x 1 e x 1 − 1
Substitution Now, let's make a substitution to simplify the limit. Let u = x 1 . As x → ∞ , u → 0 . So the limit becomes u → 0 lim 14 u e u − 1
Applying L'Hospital's Rule We can see that this limit is in the indeterminate form 0 0 , since e 0 − 1 = 1 − 1 = 0 and u approaches 0. Therefore, we can apply L'Hospital's rule, which states that if lim x → c g ( x ) f ( x ) is of the form 0 0 or ∞ ∞ , then lim x → c g ( x ) f ( x ) = lim x → c g ′ ( x ) f ′ ( x ) , provided the limit exists.
Applying L'Hospital's rule, we differentiate the numerator and the denominator with respect to u :
u → 0 lim 14 u e u − 1 = 14 u → 0 lim d u d ( u ) d u d ( e u − 1 ) = 14 u → 0 lim 1 e u
Evaluating the Limit Now we evaluate the limit: 14 u → 0 lim e u = 14 ⋅ e 0 = 14 ⋅ 1 = 14
Final Answer Therefore, the limit is 14.
Examples
L'Hospital's rule is a powerful tool used in various fields like physics and engineering to evaluate limits of indeterminate forms. For instance, in circuit analysis, when analyzing the behavior of circuits as certain parameters approach extreme values (like time approaching infinity), L'Hospital's rule can help determine the steady-state behavior of the circuit. Similarly, in fluid dynamics, it can be used to analyze the behavior of fluid flow near singularities or boundaries. Understanding and applying L'Hospital's rule allows engineers and scientists to accurately model and predict the behavior of complex systems.
