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Longhand

Chapter 3: Applications of derivatives

L’Hôpital’s rule

L’Hôpital’s rule turns a limit of the form 0/0 or ∞/∞ into a limit of derivatives. The rule is one line; the marks are in checking that the form really is indeterminate before using it, in rewriting a product, difference or power so that it is, and in stopping as soon as the form is no longer indeterminate.

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Practice questions

Every question below is original material written for Longhand. Open a solution to read the full working, or answer it yourself to have it counted towards your progress.

  • Introductory1 mark · 2 min

    L'Hôpital's rule may only be applied to a limit that is genuinely indeterminate. Which of the following limits is of the form 00\tfrac{0}{0} or ∞∞\tfrac{\infty}{\infty}?

    Answer this question yourself
  • Introductory1 mark · 2 min

    Evaluate the limit.

    lim⁡x→1ln⁡ ⁣(x2)x−1\lim_{x \to 1} \frac{\ln\!\left(x^{2}\right)}{x - 1}
    Answer this question yourself
  • Introductory2 marks · 3 min

    Evaluate the limit.

    lim⁡x→0e3x−1sin⁡(2x)\lim_{x \to 0} \frac{e^{3x} - 1}{\sin(2x)}
    Answer this question yourself
  • Standard2 marks · 4 min

    Evaluate the limit.

    lim⁡x→01−cos⁡(3x)x2\lim_{x \to 0} \frac{1 - \cos(3x)}{x^{2}}
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  • Standard2 marks · 3 min

    Evaluate lim⁡x→0+x2ln⁡x\displaystyle\lim_{x \to 0^{+}} x^{2}\ln x.

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  • Standard2 marks · 3 min

    Evaluate the limit.

    lim⁡x→0sin⁡(5x)e2x−1\lim_{x \to 0} \frac{\sin(5x)}{e^{2x} - 1}
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  • Standard3 marks · 5 min

    Evaluate the limit.

    lim⁡x→1(1ln⁡x−1x−1)\lim_{x \to 1} \left(\frac{1}{\ln x} - \frac{1}{x - 1}\right)
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  • Challenging3 marks · 5 min

    Evaluate the limit.

    lim⁡x→∞(1+3x)x\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{x}
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  • Challenging4 marks · 6 min

    There is exactly one value of the constant cc for which the limit below exists. Find that cc, and enter the value of the limit.

    lim⁡x→0e2x−1+cxx2\lim_{x \to 0} \frac{e^{2x} - 1 + cx}{x^{2}}
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  • Capstone5 marks · 8 min

    Evaluate the limit.

    lim⁡x→0cos⁡(2x)−1+2x2x4\lim_{x \to 0} \frac{\cos(2x) - 1 + 2x^{2}}{x^{4}}
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