Fundamental Limits of Calculus

Summary

A small set of standard limits underpins derivatives of trigonometric and exponential functions. The three below are the ones used most often in first-year calculus.

Prerequisites

Limits, Trigonometry, Exponentials and Logarithms

Main Result / Formula

Assume angles are in radians.

Sine over argument

lim_{x \to 0} \frac{\sin x}{x} = 1.

Cosine deficit

lim_{x \to 0} \frac{1 - \cos x}{x} = 0, lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = \frac{1}{2} .

Exponential / continuous compounding

lim_{x \to \infty} {(1 + \frac{1}{x})}^{x} = e, lim_{h \to 0} \frac{e^{h} - 1}{h} = 1.

Conditions / Assumptions

Limits involving $\sin$ and $\cos$ require radian measure.
The squeeze theorem is the usual rigorous path for $\sin x / x$ .
The base $e$ may be defined via one of the exponential limits above; then the others become theorems.

Worked Example

lim_{x \to 0} \frac{\sin (5 x)}{x} = lim_{x \to 0} 5 \cdot \frac{\sin (5 x)}{5 x} = 5 \cdot 1 = 5.

Common Mistakes

Evaluating $\sin x / x$ in degree mode.
Replacing $\sin x$ by $x$ without justification outside a limit $x \to 0$ .

Connections

Related: Squeeze Theorem, Derivatives
Next: derivatives of $\sin$ , $\cos$ , $e^{x}$

References

These standard limits appear throughout calculus texts.^[1]

OpenStax, Calculus Volume 1, https://openstax.org/details/books/calculus-volume-1 ↩︎