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Longhand

Before the midterm

Cancelling terms instead of factors

The habit: Only something that multiplies the whole top and the whole bottom can be struck out; a piece that is added cannot.

4 practice questions are built to catch it, in 4 topics.

Practise against it: 3 questions

What it looks like

A question from functions, graphs and inverses, with the answer this slip gives beside the right one.

Standard2 marks · 3 min

Which expression equals the one below wherever both are defined?

x2−9x2−x−6\frac{x^{2} - 9}{x^{2} - x - 6}
With the slip
9x+6\displaystyle \dfrac{9}{x + 6}
Worked through
x+3x+2\displaystyle \dfrac{x + 3}{x + 2}

Only a whole factor shared by the top and the bottom may be divided out; the x² on each side is part of a sum.

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
    A student evaluates the limit below as shown. Find the first line that is wrong. lim⁡x→2x2−4x2−3x+2\displaystyle \lim_{x \to 2} \frac{x^{2} - 4}{x^{2} - 3x + 2}

    LimitsStandard

  2. 02
    Two students start on the limit below. Student P applies l’Hôpital’s rule, and student Q divides the top and the bottom by 3x3^{x}. Which statement is right? lim⁡x→∞4⋅3x+2x3x+5⋅2x\displaystyle \lim_{x \to \infty} \frac{4\cdot 3^{x} + 2^{x}}{3^{x} + 5\cdot 2^{x}}

    L’Hôpital’s ruleChallenging

  3. 03
    Find the constants aa and bb that make ff continuous everywhere, and enter the product abab. Here log⁡\log is the natural logarithm.

    ContinuityCapstone

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