Evaluate the limit using L'Hôpital's rule [tex]$\lim _{x \rightarrow 0} \frac{e^x-x-1}{7 x^2}$[/tex]
Answer (2)
Using L'Hôpital's rule, we evaluated the limit lim x → 0 7 x 2 e x − x − 1 by first confirming it was in an indeterminate form of 0 0 . After applying the rule twice, we found the limit is 14 1 .
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Recognize the limit is in 0 0 indeterminate form.
Apply L'Hôpital's rule once, differentiating the numerator and denominator to get lim x → 0 14 x e x − 1 .
Apply L'Hôpital's rule again, resulting in lim x → 0 14 e x .
Evaluate the limit to find the answer: 14 1 .
Explanation
Problem Setup We are asked to evaluate the limit lim x → 0 7 x 2 e x − x − 1 using L'Hôpital's rule.
Check Indeterminate Form First, we check if we can directly substitute x = 0 into the expression. We have 7 ( 0 ) 2 e 0 − 0 − 1 = 0 1 − 0 − 1 = 0 0 . Since we have an indeterminate form of type 0 0 , we can apply L'Hôpital's rule.
First Application of L'Hôpital's Rule Applying L'Hôpital's rule once, we differentiate the numerator and the denominator with respect to x . The derivative of the numerator e x − x − 1 is e x − 1 . The derivative of the denominator 7 x 2 is 14 x . Thus, the limit becomes lim x → 0 14 x e x − 1 .
Check Indeterminate Form Again Now, we check if we can directly substitute x = 0 into the new expression. We have 14 ( 0 ) e 0 − 1 = 0 1 − 1 = 0 0 . Since we still have an indeterminate form of type 0 0 , we can apply L'Hôpital's rule again.
Second Application of L'Hôpital's Rule Applying L'Hôpital's rule a second time, we differentiate the numerator and the denominator with respect to x . The derivative of the numerator e x − 1 is e x . The derivative of the denominator 14 x is 14 . Thus, the limit becomes lim x → 0 14 e x .
Evaluate the Limit Now, we evaluate the limit by substituting x = 0 into the expression. We have 14 e 0 = 14 1 .
Final Answer Therefore, the limit is 14 1 .
Examples
L'Hôpital's rule is often used in physics and engineering to simplify calculations involving indeterminate forms. For example, when analyzing the behavior of circuits or fluid dynamics, you might encounter expressions that are initially undefined but can be resolved using L'Hôpital's rule to find meaningful results. This allows engineers to accurately model and predict system behavior, even when faced with complex mathematical expressions.
