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Longhand

Before Quiz 2

Applying l’Hôpital’s rule where the form is not indeterminate

The habit: Check for 0/0 or ∞/∞ before differentiating; a top and bottom heading anywhere else, such as 1 over 0 or 0 over ∞, decide the limit directly, with a sign check when the bottom tends to 0.

3 practice questions are built to catch it, in 1 topic.

See it fail in a figure: Not 0/0, in the L’Hôpital’s rule lesson

Practise against it: 2 questions

What it looks like

A question from L’Hôpital’s rule, with the answer this slip gives beside the right one.

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}?

With the slip
lim⁡x→0+1x\displaystyle \lim_{x \to 0^{+}} \dfrac{1}{x}
Worked through
lim⁡x→0ex−1x\displaystyle \lim_{x \to 0} \dfrac{e^{x} - 1}{x}

1/x as x → 0+ is 1 over a shrinking positive bottom, so it goes straight to +∞: 1/0 is not 0/0.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    Evaluate the limit. lim⁡x→1x2+log⁡xx+1\displaystyle \lim_{x \to 1} \frac{x^{2} + \log x}{x + 1}

    L’Hôpital’s ruleIntroductory

  2. 02
    Two students evaluate lim⁡x→1xlog⁡x−x+1x2−1\displaystyle\lim_{x \to 1} \frac{x\log x - x + 1}{x^{2} - 1} as shown. Which solution is right, and why?

    L’Hôpital’s ruleStandard

Start the set