Standard limits

A handful of standard limits appear over and over. Memorise them; recognise them in problems; reduce harder limits to them by algebraic manipulation.

The trig limits

(L1) $\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1.$

(Here $x$ is in radians.) This is the most important limit in elementary calculus. Geometric argument: in a unit circle, for small $x > 0$, the arc length $x$, the chord $\sin x$, and the tangent $\tan x$ satisfy $\sin x < x < \tan x$. Divide by $\sin x$: $1 < x/\sin x < 1/\cos x$. As $x \to 0$, $1/\cos x \to 1$, so $x/\sin x \to 1$, hence $\sin x / x \to 1$.

(L2) $\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}.$

Derive from L1: $\dfrac{1 - \cos x}{x^2} = \dfrac{2 \sin^2(x/2)}{x^2} = \dfrac{1}{2}\left(\dfrac{\sin(x/2)}{x/2}\right)^2 \to \dfrac{1}{2}$.

(L3) $\displaystyle \lim_{x \to 0} \frac{\tan x}{x} = 1.$ (Combine L1 and $\cos x \to 1$.)

The exponential / logarithmic limits

(L4) $\displaystyle \lim_{x \to 0} \frac{e^x - 1}{x} = 1.$

(L5) $\displaystyle \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$, for any $a > 0$. (Use $a = e^{\ln a}$.)

(L6) $\displaystyle \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1.$

(L7) $\displaystyle \lim_{x \to 0} (1 + x)^{1/x} = e \approx 2.71828\dots$

This is one definition of the number $e$. Equivalently, $\lim_{n \to \infty}\left(1 + \dfrac{1}{n}\right)^n = e$.

The polynomial limit

(L8) $\displaystyle \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n - 1}.$

This is the basic limit for differentiating $x^n$.

Strategy: recognise and reduce

When you see a limit with $\sin$, $\cos$, $e^x$, $a^x$, or $\ln$, look for one of L1–L7. Manipulate the expression , by factoring, substitution, or multiplying numerator and denominator by a clever expression , to expose the standard form.

Worked examples

Example 1. $\lim_{x \to 0} \dfrac{\sin 3 x}{x}$.

Write $\dfrac{\sin 3x}{x} = 3 \cdot \dfrac{\sin 3 x}{3 x}$. Let $u = 3 x$; as $x \to 0$, $u \to 0$. Limit = $3 \cdot 1 = 3$.

Example 2. $\lim_{x \to 0} \dfrac{\sin 5 x}{\sin 7 x}$.

$\dfrac{\sin 5x}{\sin 7 x} = \dfrac{\sin 5x / (5 x)}{\sin 7 x / (7 x)} \cdot \dfrac{5 x}{7 x} = \dfrac{\sin 5x/(5x)}{\sin 7x/(7x)} \cdot \dfrac{5}{7} \to \dfrac{1}{1} \cdot \dfrac{5}{7} = \dfrac{5}{7}$.

Example 3. $\lim_{x \to 0} \dfrac{1 - \cos 2 x}{x^2}$.

$\dfrac{1 - \cos 2 x}{x^2} = \dfrac{2 \sin^2 x}{x^2} = 2 \left(\dfrac{\sin x}{x}\right)^2 \to 2$.

Example 4. $\lim_{x \to 0} \dfrac{e^{2 x} - 1}{x}$.

$\dfrac{e^{2 x} - 1}{x} = 2 \cdot \dfrac{e^{2 x} - 1}{2 x} \to 2 \cdot 1 = 2$.

Example 5. $\lim_{x \to 0} \dfrac{\ln(1 + 3 x)}{x}$.

$\dfrac{\ln(1 + 3 x)}{x} = 3 \cdot \dfrac{\ln(1 + 3 x)}{3 x} \to 3 \cdot 1 = 3$.

Try it yourself

1. $\lim_{x \to 0} \dfrac{\sin 7 x}{x}$.
2. $\lim_{x \to 0} \dfrac{\sin 2 x}{\sin 3 x}$.
3. $\lim_{x \to 0} \dfrac{\tan 3 x}{x}$.
4. $\lim_{x \to 0} \dfrac{1 - \cos 3 x}{x^2}$.
5. $\lim_{x \to 0} \dfrac{e^{3 x} - 1}{2 x}$.
6. $\lim_{x \to 0} \dfrac{5^x - 1}{x}$.
7. $\lim_{x \to 0} \dfrac{\ln(1 + 5 x)}{2 x}$.
8. $\lim_{x \to 0} \dfrac{\sin x - \tan x}{x^3}$. (Hint: $\sin x - \tan x = -\sin x (1 - \cos x)/\cos x$.)
9. $\lim_{x \to 0} \dfrac{\sin x^2}{x}$.
10. $\lim_{x \to a} \dfrac{x^3 - a^3}{x - a}$.
11. $\lim_{x \to 0} \dfrac{\tan x - \sin x}{x^3}$.
12. $\lim_{n \to \infty}\left(1 + \dfrac{2}{n}\right)^n$. (Hint: write as $((1 + 2/n)^{n/2})^2 \to e^2$.)

Pitfalls / Tricks

• $x$ in $\sin x / x$ etc. must be in radians.
• When numerator and denominator both go to $0$, force one of the standard forms (e.g., $\sin (ax)/x = a \cdot \sin(ax)/(ax)$).
• For $(1 + f(x))^{g(x)}$ with $f(x) \to 0$ and $g(x) \to \infty$, often write as $e^{g(x) \ln(1 + f(x))}$ and use L6.
• Insight. Standard limits are the vocabulary of calculus. Once you spot them, almost every $0/0$ problem opens up.