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Worksheet
L'Hôpital's Rule
IB Mathematics AA · HL · Shadow Worksheet
Practice
Name
Question 1
Use L'Hôpital's rule to evaluate the following limits:
(a)\(\displaystyle\lim_{x o0}\dfrac{\sin x}{x}\)
(b)\(\displaystyle\lim_{x o1}\dfrac{x^3-1}{x-1}\)
(c)\(\displaystyle\lim_{x o0}\dfrac{e^x-1}{x}\)
(d)\(\displaystyle\lim_{x o\infty}\dfrac{x^2}{e^x}\)
(a)
\( frac{0}{0}\) form; apply L'H
\(\dfrac{\cos x}{1}\Big|_{x=0}\)
Answer
\(1\)
(b)
\( frac{0}{0}\) form
\(\dfrac{3x^2}{1}\Big|_{x=1}\)
Answer
\(3\)
(c)
\( frac{0}{0}\) form
\(\dfrac{e^x}{1}\Big|_{x=0}\)
Answer
\(1\)
(d)
\( frac{\infty}{\infty}\): apply twice
\(\dfrac{2x}{e^x} o\dfrac{2}{e^x} o0\)
Answer
\(0\)
Question 2
Evaluate \(\displaystyle\lim_{x o0}\dfrac{\sin(3x)}{x\cos(2x)}\).
Apply L'H
\(\dfrac{3\cos(3x)}{\cos(2x)-2x\sin(2x)}\Big|_{x=0}=\dfrac{3}{1}\)
Answer
\(3\)
Question 3
Evaluate \(\displaystyle\lim_{x o0^+}x\ln x\).
Rewrite as fraction
\(\lim_{x o0^+}\dfrac{\ln x}{1/x}\); \( frac{-\infty}{\infty}\) form; apply L'H: \(\dfrac{1/x}{-1/x^2}=-x o0\)
Answer
\(0\)
Question 4
Evaluate \(\displaystyle\lim_{x o0}\dfrac{1-\cos x}{x^2}\).
Apply L'H twice
\(\dfrac{\sin x}{2x} o\dfrac{\cos x}{2}\Big|_{x=0}\)
Answer
\(\dfrac{1}{2}\)
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