Proofs of all Limit Formulas

Limit Formulas with Their Proofs:

The limit of a function is an important concept in the theory of Calculus. In this section, we will prove all the limit formulas

Formula 1:  `lim_{x to 0} frac{sin x}{x}=1`

Formula 2:  `lim_{x to a} frac{x^n-a^n}{x-a}=na^{n-1}`

Formula 3:  `lim_{x to 0} frac{e^x-1}{x}=1`

Formula 4:  `lim_{x to 0} frac{log_e(1+x)}{x}=1`

Formula 5:  `lim_{x to 0} frac{a^x-1}{x}=log_e a (a>0)`

Formula 6:  `lim_{x to 0} (1+x)^{frac{1}{x}}=e`

Formula 7:  `lim_{x to infty} (1+frac{1}{x})^x=e`

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