Use L'Hospital to determine the following limit. Use exact values.
[tex]\lim _{x \rightarrow 0^{+}}(1+\sin (5 x))^{\frac{1}{x}}=\square[/tex]
Answer (2)
By applying L'Hospital's rule on the limit lim x → 0 + ( 1 + sin ( 5 x ) ) x 1 , we transformed it into the logarithmic form, recognized it as an indeterminate form, and differentiated to find that the limit evaluates to e 5 .
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Recognize the limit as an indeterminate form of type 0 0 after taking the natural logarithm.
Apply L'Hopital's rule by differentiating the numerator and denominator of the logarithmic expression.
Evaluate the resulting limit as x approaches 0 .
Exponentiate the result to find the original limit: e 5 .
Explanation
Problem Analysis We are asked to find the limit of the function ( 1 + sin ( 5 x ) ) x 1 as x approaches 0 from the right. We are instructed to use L'Hospital's rule and find the exact value of the limit.
Taking the Natural Logarithm Let L = lim x → 0 + ( 1 + sin ( 5 x ) ) x 1 . To apply L'Hospital's rule, we first need to transform the expression into a fraction. We take the natural logarithm of both sides: ln L = lim x → 0 + x 1 ln ( 1 + sin ( 5 x )) = lim x → 0 + x l n ( 1 + s i n ( 5 x )) .
Indeterminate Form As x → 0 + , ln ( 1 + sin ( 5 x )) → ln ( 1 + 0 ) = 0 and x → 0 . Thus, we have an indeterminate form of type 0 0 , which allows us to apply L'Hospital's rule.
Applying L'Hospital's Rule Applying L'Hospital's rule, we differentiate the numerator and the denominator with respect to x : ln L = lim x → 0 + d x d x d x d l n ( 1 + s i n ( 5 x )) = lim x → 0 + 1 1 + s i n ( 5 x ) 5 c o s ( 5 x ) = lim x → 0 + 1 + s i n ( 5 x ) 5 c o s ( 5 x ) .
Evaluating the Limit Now we evaluate the limit as x approaches 0 : ln L = 1 + s i n ( 0 ) 5 c o s ( 0 ) = 1 + 0 5 ( 1 ) = 5 .
Solving for L Finally, we solve for L by taking the exponential of both sides: L = e l n L = e 5 .
Final Answer Therefore, the limit is e 5 .
Examples
In thermodynamics, the behavior of systems approaching equilibrium can be modeled using limits similar to the one we just solved. For instance, the rate of a chemical reaction approaching equilibrium might be described by an expression involving a limit. Understanding how to evaluate such limits, especially using tools like L'Hopital's rule, helps predict the final state and rate of these reactions, crucial for designing efficient chemical processes and understanding natural phenomena.
