# Limit Formulas

Limit and continuity Formulas

Concept of limit and continuity was developed in 17th century by mathematicians, primarily to foster the development of calculas. The concept of the limit is very important in terms of calculas.

Some important formulas of limit and continuity are as follows:-

1. $\displaystyle{\lim_{x\to a} \frac{x^n- a^n}{x-a} = na^{n- 1}}$, for all rational value of n
2. $\displaystyle{\lim_{x \to 0}\frac{\sin x}{x} = 1}$ or $\displaystyle{\lim_{x \to 0}\frac{x}{\sin x} = 1} \\[5mm]$ $3. \displaystyle{\lim_{x \to 0} \frac{e^x- 1}{x} = 1} \\[5mm] 4. \displaystyle{\lim_{x\ to 0} \frac{\log (1 + x)}{x} = 1} \\[5mm] 5. \displaystyle{\lim_{x \to \infty} \left( 1 + \dfrac{1}{x} \right)^n = e} \\[5mm] 6. \displaystyle{\lim_{x \to 0}(1 + x)^\frac{1}{x} = e} \\[5mm] 7. \displaystyle{\lim_{x \to 0}\frac{\tan x}{x} = 1}$

Note: indeterminate forms are $\dfrac{0}{0}, \infty^0, \dfrac{\infty}{\infty}, \infty- \infty, 1^\infty, 0^\infty$

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