Limits & Standard Forms
Formula Sheet 7
1. Trigonometric & Inverse Limits
| Standard Form | Result |
|---|---|
| $$ \lim_{x \to 0} \frac{\sin x}{x} $$ | $$ 1 $$ |
| $$ \lim_{x \to 0} \frac{\tan x}{x} $$ | $$ 1 $$ |
| Inverse Sine: $$ \lim_{x \to 0} \frac{\sin^{-1} x}{x} $$ | $$ 1 $$ |
| Inverse Tan: $$ \lim_{x \to 0} \frac{\tan^{-1} x}{x} $$ | $$ 1 $$ |
2. Algebraic Standard Limit
$$ \lim_{x \to a} \frac{x^n – a^n}{x – a} = n \cdot a^{n-1} $$
(Where ‘n’ is any rational number)
3. Exponential & Logarithmic
Exponential ($$ e^x $$)
$$ \lim_{x \to 0} \frac{e^x – 1}{x} = 1 $$
General Base ($$ a^x $$)
$$ \lim_{x \to 0} \frac{a^x – 1}{x} = \log_e a $$
Logarithmic
$$ \lim_{x \to 0} \frac{\log_e(1+x)}{x} = 1 $$