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`