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Longhand

Chapter 1: Limits

Continuity and the Intermediate Value Theorem

A function is continuous at a point when its limit there is its value there. Most marks in this topic come from applying that definition to a piecewise function with an unknown constant, and from the Intermediate Value Theorem, the first result in the course that lets you prove something exists without finding it.

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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

    A function ff is continuous at x=2x = 2, and lim⁡x→2f(x)=5\lim_{x \to 2} f(x) = 5. Which of the following must be true?

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  • Introductory1 mark · 2 min

    The function below is undefined at x=3x = 3. Find the value kk that should be assigned to f(3)f(3) so that ff is continuous there.

    f(x)=x2−9x−3,x≠3f(x) = \frac{x^{2} - 9}{x - 3}, \qquad x \neq 3
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  • Introductory2 marks · 3 min

    On which set is g(x)=1x2−4g(x) = \dfrac{1}{\sqrt{x^{2} - 4}} continuous?

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  • Standard2 marks · 3 min

    Find the value of aa that makes ff continuous at x=1x = 1.

    f(x)={2x+1,x<1ax2−3,x≥1f(x) = \begin{cases} 2x + 1, & x < 1 \\[4pt] a x^{2} - 3, & x \geq 1 \end{cases}
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  • Standard2 marks · 4 min

    Find the value of the constant cc that makes ff continuous everywhere.

    f(x)={cx2−1,x<23x+c,x≥2f(x) = \begin{cases} cx^{2} - 1, & x < 2 \\[4pt] 3x + c, & x \geq 2 \end{cases}
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  • Standard3 marks · 5 min

    Use the Intermediate Value Theorem, together with the fact that a cubic has at most three real roots, to determine exactly how many real roots the equation below has.

    x3−4x+1=0x^{3} - 4x + 1 = 0
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  • Standard2 marks · 3 min

    Classify the discontinuities of the function below.

    h(x)=x2−x−2x2−4h(x) = \frac{x^{2} - x - 2}{x^{2} - 4}
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  • Challenging3 marks · 5 min

    Find the value of kk that makes ff continuous at x=0x = 0.

    f(x)={x+4−2x,x≠0k,x=0f(x) = \begin{cases} \dfrac{\sqrt{x + 4} - 2}{x}, & x \neq 0 \\[8pt] k, & x = 0 \end{cases}
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  • Challenging4 marks · 6 min

    The function below is continuous everywhere. Find a+ba + b.

    f(x)={ax+3,x<1x2+b,1≤x<32x−1,x≥3f(x) = \begin{cases} ax + 3, & x < 1 \\[4pt] x^{2} + b, & 1 \leq x < 3 \\[4pt] 2x - 1, & x \geq 3 \end{cases}
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  • Capstone5 marks · 10 min

    Show that the equation cos⁡x=x2\cos x = x^{2} has at least two real solutions.

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