AP Calculus BC 4.7 L Hopital Rule Overview

Review L Hopital questions by recognising indeterminate forms and simplifying limits through repeated differentiation while checking that the rule applies.

Covered syllabus points

LIM-4.A—Determine limits of functions that result in indeterminate forms

[Maximum number: 3]

Consider the differential equation $\frac{d y}{d x}=x^{2}-\frac{1}{2} y$ .

Let y=g(x) be the particular solution to the given differential equation with g(-1)=2. Find $\lim _{x \rightarrow-1}\left(\frac{g(x)-2}{3(x+1)^{2}}\right)$ . Show the work that leads to your answer.

$\lim _{x \rightarrow-1}(g(x)-2)=0$ and $\lim _{x \rightarrow-1} 3(x+1)^{2}=0$
Using L'Hospital's Rule,

\lim _{x \rightarrow-1}\left(\frac{g(x)-2}{3(x+1)^{2}}\right)=\lim _{x \rightarrow-1}\left(\frac{g^{\prime}(x)}{6(x+1)}\right)

$\lim _{x \rightarrow-1} g^{\prime}(x)=0$ and $\lim _{x \rightarrow-1} 6(x+1)=0$
Using L'Hospital's Rule,

\lim _{x \rightarrow-1}\left(\frac{g^{\prime}(x)}{6(x+1)}\right)=\lim _{x \rightarrow-1}\left(\frac{g^{\prime \prime}(x)}{6}\right)=\frac{-2}{6}=-\frac{1}{3}

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Covered syllabus points

4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms question 1