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Longhand

Before the midterm

Approaching from the wrong side

The habit: For a one-sided limit, check the sign each factor takes on that side.

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

Practise against it: 6 questions

What it looks like

A question from continuity, with the answer this slip gives beside the right one.

Introductory2 marks · 2 min

The graph of a function ff is shown. Which statement about ff at x=2x = 2 is true?

The graph of y = f(x) for x from −1 to 5, in two pieces. The left piece is a shallow upward bowl: it starts at about (−1, 2.25), dips to (0, 2) and rises to an open circle at (2, 3). The right piece is a straight line that starts at a filled point at (2, 1) and rises to (5, 4).-112345-112345
With the slip
The limit as x → 2 is 3, the height of the open circle, and f is not continuous at x = 2 because f(2) = 1.
Worked through
The left-hand limit is 3 and the right-hand limit is 1, so the limit as x → 2 does not exist and f is not continuous there.

Only the left piece runs into the open circle at height 3. A two-sided limit needs the right piece to head there too: check where it starts.

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 one-sided limit. lim⁡x→4+x+24−x\displaystyle \lim_{x \to 4^{+}} \frac{x + 2}{4 - x}

    LimitsIntroductory

  2. 02
    Which statement about the limit of the function below as x→3x \to 3 is correct? f(x)=x2−x−6∣x−3∣\displaystyle f(x) = \frac{x^{2} - x - 6}{|x - 3|}

    LimitsIntroductory

  3. 03
    The graph of a function hh is shown, with dashed asymptotes at x=−2x = -2 and y=1y = 1. Use it to find the one-sided limit below. lim⁡x→−2+h(x)\displaystyle \lim_{x \to -2^{+}} h(x)

    ContinuityIntroductory

  4. 04
    Evaluate the one-sided limit. lim⁡x→3−∣x−3∣x2−9\displaystyle \lim_{x \to 3^{-}} \frac{|x - 3|}{x^{2} - 9}

    LimitsStandard

  5. 05
    The graph of a function ff is shown. Of x=−3x = -3, x=−1x = -1, x=1x = 1 and x=3x = 3, which set lists exactly the ones at which ff is not continuous?

    ContinuityStandard

  6. 06
    Two students evaluate the one-sided limit lim⁡x→−2+x2+x−2x2+4x+4\displaystyle\lim_{x \to -2^{+}} \frac{x^{2} + x - 2}{x^{2} + 4x + 4} as shown. Which solution is right, and why?

    LimitsChallenging

Start the set