Computing limits
Limits by substitution and cancelling, limits of fractions at infinity and the limit of sin t / t. The forms 0/0 and ∞/∞ — and how to get rid of them.
How to solve it
A limit is the value a function approaches as $x$ approaches a point. If substitution gives a number, that is the answer. If it gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, the expression has to be transformed — most often by cancelling a common factor or dividing by the highest power.
Step by step
- Substitute the point. A number — done.
- $\frac{0}{0}$ in a fraction of polynomials: both the top and the bottom are divisible by $x - a$. Factor, cancel, substitute again.
- $x \to \infty$: divide the top and the bottom by the highest power of $x$. Terms like $\frac{c}{x^k}$ go to zero, and the ratio of the leading coefficients is left.
- With a sine: fit the expression to $\frac{\sin t}{t} \to 1$ by multiplying and dividing by the right number.
Common mistakes
- Seeing $\frac{0}{0}$ and answering “$0$” or “no limit”. It is a signal to transform the expression, not an answer.
- “Cancelling the sine” in $\frac{\sin 3x}{5x}$ to get $\frac{3}{5}$ — right by accident; the honest way is the special limit with a factor of $3$.
- At infinity comparing the constant terms instead of the highest powers.
- Cancelling $x - a$ and forgetting to substitute the point again.