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Longhand

MATH 100 limits practice

A limit describes where a function is heading as its input approaches some value, which is not always where the function arrives — and sometimes the function never arrives at all. Almost everything that follows in MATH 100 is defined in terms of one, so the fluency you build here is spent for the rest of the term.

Most limit questions on a first-term paper fall into four shapes: substitution works immediately, substitution gives 0/0 and the expression needs rewriting, the two sides disagree, or the variable runs off to infinity. Recognising which shape you are looking at is most of the work; the algebra afterwards is usually short.

The questions below cover all four, from a factor-and-cancel warm-up to a squeeze theorem argument and a continuity problem with an unknown constant.

Begin practice sessionBrowse all 7 questions

What you should be able to do

  • Decide whether substitution is enough

    For a function continuous at the point, the limit is just the value. Knowing when you are allowed to stop there saves time you will want later in the paper.

  • Resolve the indeterminate form 0/0

    Factor and cancel, multiply by a conjugate, or combine fractions over a common denominator. Each technique targets a different reason the zero appeared.

  • Work with one-sided limits

    Evaluate from the left and the right separately, and state that the two-sided limit exists only when they agree. Piecewise functions and vertical asymptotes both depend on this.

  • Handle limits at infinity

    Divide through by the highest power present and let every reciprocal term vanish. For rational functions this reduces to comparing leading terms, sign included.

  • Apply the squeeze theorem

    When a factor oscillates without settling, bound it, multiply the inequality through by something that tends to zero, and trap the limit between two known ones.

  • State continuity in terms of limits

    A function is continuous at a point when the limit exists, the value exists, and the two are equal. Questions asking you to choose a constant are testing exactly this definition.

Where marks are usually lost

  • Reading 0/0 as "the limit does not exist"

    An indeterminate form is a signal that the expression has not been simplified, not a verdict. The function is genuinely undefined at the point; the limit near it may still be a perfectly ordinary number.

  • Splitting a root across a sum

    The square root of a sum is not the sum of the square roots. When a root is causing the trouble, the tool is the conjugate, not distribution.

  • Losing a sign in a limit at infinity

    In an expression such as (3x² − 5x)/(7 − 2x²) the leading coefficient of the denominator is −2, not 2. Rewriting the denominator in descending order before comparing prevents this.

  • Applying sin(u)/u when the arguments do not match

    The standard limit equals 1 only when the argument of the sine and the denominator are identical. For sin(3x)/(5x) the answer is 3/5, reached by forcing the match first.

  • Answering an unsigned infinity at a vertical asymptote

    When the two sides run to +∞ and −∞ the two-sided limit does not exist, and the marks are for saying which side does which. A single unsigned answer cannot be right.

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

    Evaluate the limit.

    lim⁡x→3x2−9x−3\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}
    Answer this question yourself
  • Standard2 marks · 3 min

    Evaluate the limit.

    lim⁡x→0x+4−2x\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}
    Answer this question yourself
  • 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}
    Answer this question yourself
  • Challenging3 marks · 5 min

    Evaluate the limit.

    lim⁡x→0x2cos⁡ ⁣(5x)\lim_{x \to 0} x^{2} \cos\!\left(\frac{5}{x}\right)
    Answer this question yourself
  • Standard2 marks · 4 min

    Evaluate the limit.

    lim⁡x→∞3x2−5x+17−2x2\lim_{x \to \infty} \frac{3x^{2} - 5x + 1}{7 - 2x^{2}}
    Answer this question yourself
  • Challenging4 marks · 6 min

    Determine both one-sided limits below, and state with justification whether the two-sided limit lim⁡x→2x−5x−2\lim_{x \to 2} \frac{x - 5}{x - 2} exists.

    lim⁡x→2−x−5x−2andlim⁡x→2+x−5x−2\lim_{x \to 2^{-}} \frac{x - 5}{x - 2} \qquad \text{and} \qquad \lim_{x \to 2^{+}} \frac{x - 5}{x - 2}
    Answer this question yourself
  • Standard2 marks · 3 min

    Evaluate the limit.

    lim⁡x→0sin⁡(3x)5x\lim_{x \to 0} \frac{\sin(3x)}{5x}
    Answer this question yourself

Common questions

Do I need to use the epsilon-delta definition in MATH 100?
Most first-term assessments ask you to evaluate limits and reason about one-sided behaviour rather than construct a formal epsilon-delta proof. Check your own course outline, because the emphasis varies between sections and years.
When can I use L’Hôpital’s rule?
Only once it has been covered, and only on an indeterminate form such as 0/0 or infinity over infinity. Early in the course the expected method is algebraic — factor, rationalise, or divide by the highest power — so a solution that leans on the rule may not earn the marks.
How much working should I show for a limit question?
Enough that each step follows from the one above. Naming the indeterminate form, showing the rewrite that removes it, and then substituting is normally the full expectation for a two or three mark question.

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