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Longhand

Before the midterm

Misjudging an asymptote or a hole

The habit: Cancel first: a vertical asymptote needs the denominator to vanish afterwards, a zero of the numerator is an intercept only where f itself is defined, and only the leading terms matter at infinity.

14 practice questions are built to catch it, in 3 topics.

Practise against it: 6 questions

What it looks like

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

Introductory2 marks · 2 min

Find the horizontal asymptote of the graph of the function below: enter the value of yy it approaches as x→±∞x \to \pm\infty.

f(x)=(1−4x)(x+3)6x2+5f(x) = \frac{(1 - 4x)(x + 3)}{6x^{2} + 5}
With the slip
0.6\displaystyle 0.6
Worked through
y=−23\displaystyle y = -\frac{2}{3}

3/5 is the ratio of the constants, 1 × 3 over 5, which is f(0). Far out the constants are negligible and the x² terms decide.

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
    Which statement gives every vertical asymptote of the graph of the function below? f(x)=x2+x−6x2+4x−12\displaystyle f(x) = \frac{x^{2} + x - 6}{x^{2} + 4x - 12}

    LimitsIntroductory

  2. 02
    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

  3. 03
    Which list gives every horizontal and vertical asymptote of the graph of the function below? f(x)=3ex+2ex−1\displaystyle f(x) = \frac{3e^{x} + 2}{e^{x} - 1}

    LimitsStandard

  4. 04
    Classify the discontinuities of the function below. h(x)=x2−x−2x2−4\displaystyle h(x) = \frac{x^{2} - x - 2}{x^{2} - 4}

    ContinuityStandard

  5. 05
    Which statement about the vertical asymptotes of the graph of the function below is correct? g(x)=x2−4x+7−3\displaystyle g(x) = \frac{x^{2} - 4}{\sqrt{x + 7} - 3}

    LimitsChallenging

  6. 06
    The function below has a horizontal asymptote as x→∞x \to \infty and another as x→−∞x \to -\infty. Find the xx-coordinate of the point where its graph crosses one of these asymptotes, or type none if it never does. f(x)=4x+3x2+1\displaystyle f(x) = \frac{4x + 3}{\sqrt{x^{2} + 1}}

    LimitsChallenging

Start the set