Limits

Key limits

Sine over x

{lim}_{x \to 0} \frac{\sin x}{x} = 1

One minus cosine over x

{lim}_{x \to 0} \frac{1 - \cos x}{x} = 0

The number e

{lim}_{x \to \infty} {(1 + \frac{1}{x})}^{x} = e

Exponential

{lim}_{x \to 0} \frac{e^{x} - 1}{x} = 1

L'Hôpital's rule

{lim}_{x \to c} \frac{f (x)}{g (x)} = {lim}_{x \to c} \frac{f^{'} (x)}{g^{'} (x)}

for 0/0 or ∞/∞ forms

Derivative as a limit

f^{'} (x) = {lim}_{h \to 0} \frac{f (x + h) - f (x)}{h}

Derivative at a point

f^{'} (c) = {lim}_{x \to c} \frac{f (x) - f (c)}{x - c}

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