Evaluate the following limit using L'Hopital's rule.
[tex]\lim _{x \rightarrow 0} \frac{\sqrt{x+1}-\left(\frac{x}{2}+1\right)}{x^2}=-\frac{[?]}{[]}[/tex]
Answer (2)
Check that the limit is in 0 0 indeterminate form.
Apply L'Hopital's rule once, obtaining lim x → 0 2 x 2 x + 1 1 − 2 1 .
Apply L'Hopital's rule again, obtaining lim x → 0 2 − 4 ( x + 1 ) 3/2 1 .
Evaluate the limit to find the answer: − 8 1 .
Explanation
Problem Setup We are asked to evaluate the limit lim x → 0 x 2 x + 1 − ( 2 x + 1 ) . We are instructed to use L'Hopital's rule.
Indeterminate Form Check if the limit is in indeterminate form. As x → 0 , the numerator approaches 0 + 1 − ( 0/2 + 1 ) = 1 − 1 = 0 , and the denominator approaches 0 2 = 0 . Thus, the limit is in the indeterminate form 0 0 , and we can apply L'Hopital's rule.
First Application of L'Hopital's Rule Apply L'Hopital's rule once by differentiating the numerator and the denominator with respect to x . The derivative of the numerator is 2 x + 1 1 − 2 1 , and the derivative of the denominator is 2 x . The new limit is lim x → 0 2 x 2 x + 1 1 − 2 1 .
Indeterminate Form Again Check if the new limit is in indeterminate form. As x → 0 , the numerator approaches 2 0 + 1 1 − 2 1 = 2 1 − 2 1 = 0 , and the denominator approaches 2 ( 0 ) = 0 . Thus, the limit is again in the indeterminate form 0 0 , and we can apply L'Hopital's rule again.
Second Application of L'Hopital's Rule Apply L'Hopital's rule a second time. Differentiate the numerator and the denominator with respect to x . The derivative of the numerator is − 4 ( x + 1 ) 3/2 1 , and the derivative of the denominator is 2 . The new limit is lim x → 0 2 − 4 ( x + 1 ) 3/2 1 .
Evaluating the Limit Evaluate the limit. As x → 0 , the expression becomes 2 − 4 ( 0 + 1 ) 3/2 1 = 2 − 4 1 = − 8 1 .
Final Answer Therefore, lim x → 0 x 2 x + 1 − ( 2 x + 1 ) = − 8 1 . The question asks for the answer in the form − [ ] [ ?] . Thus, the answer is − 8 1 .
Examples
In physics, when analyzing the motion of objects under certain forces, you might encounter limits similar to this one. For example, when studying damped oscillations or the behavior of circuits as they approach a steady state, evaluating such limits helps determine the system's long-term behavior. L'Hopital's rule becomes a handy tool to simplify these calculations and understand the system's response as certain parameters approach specific values.
Using L'Hopital's rule, the limit lim x → 0 x 2 x + 1 − ( 2 x + 1 ) evaluates to − 8 1 . This limit is approached by confirming an indeterminate form and applying L'Hopital's rule twice. The final result is − 8 1 .
;
