Lesson

Limits

A limit describes the value a function approaches as its input approaches some point. For a continuous function you can often evaluate a limit by direct substitution; some limits need a little algebra first.

Practice

Q1

Evaluate $\displaystyle\lim_{x \to 2} x^2$.

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$2^2$.

Q2

Evaluate $\displaystyle\lim_{x \to 3} (2x + 1)$.

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Substitute $x = 3$ directly.

Some limits give the indeterminate form $\\tfrac{0}{0}$ on substitution. Factor and cancel, or use a known standard limit, before evaluating.

Quiz

Q1

Evaluate $\displaystyle\lim_{x \to 0} \frac{\sin x}{x}$.

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A standard limit worth memorising.

Q2

Evaluate $\displaystyle\lim_{x \to \infty} \frac{1}{x}$.

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As $x$ grows, $1/x$ shrinks toward this value.

Q3

Evaluate $\displaystyle\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$.

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Factor the top and cancel $(x-1)$.