Limits using Maclaurin series and L'Hopital's ruleAQA A-Level Further Maths: Flashcards

What these 13 flashcards ask

State l'Hôpital's rule.
Is l'Hôpital's rule the quotient rule?
What are the two forms for which l'Hôpital's rule applies directly?
What if the result of l'Hôpital's rule is still \frac00?
\lim{x\to0}\frac{\sin x}{x}?
\lim{x\to0}\frac{1-\cos x}{x^2}?
\lim{x\to0}\frac{e^x-1}{x}?
\lim{x\to\infty}\frac{x^k}{e^x}?
\lim{x\to\infty}\frac{\ln x}{x}?
How do you find a limit at 0 using series?
Why is 0\times\infty not a valid answer?
\lim{x\to0^+}x\ln x?
How many series terms should you take for a limit?

Exam questions on Limits using Maclaurin series and L'Hopital's rule

Let $L=\lim_{x\to0}\dfrac{1-\cos2x}{x^2}$ .
Use the Maclaurin series for $\cos x$ to find the value of $L$ .2 marks
Let $M=\lim_{x\to\infty}\dfrac{x^2}{\mathrm{e}^{x}}$ .
Show how applying l'Hôpital's rule twice gives your answer to (b).2 marks
Let $F(x)=\dfrac{\sin x-x\cos x}{x^3}$ for $x\ne0$ .
Use Maclaurin series to find $\lim_{x\to0}F(x)$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).