Limits of Algebraic functions

There are six limit rules for the algebraic functions in calculus and they’re used as formulas to find the limits of algebraic functions. One limit rule is purely in algebraic form, another one rule is in logarithmic form and remaining formulas are in exponential form.

$(1) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{x^n-a^n}{x-a}}$ $\,=\,$ $na^{n-1}$

Logarithmic function

$(2) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\log_{e}{(1+x)}}{x}}$ $\,=\,$ $1$

Exponential functions

$(3) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^x-1}{x}}$ $\,=\,$ $\log_{e}{a}$

$(4) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{e^x-1}{x}}$ $\,=\,$ $1$

$(5) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize {(1+x)}^{\frac{1}{x}}}$ $\,=\,$ $e$

$(6) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize {\Bigg(1+\dfrac{1}{x}\Bigg)}^x}$ $\,=\,$ $e$

Problems

List of solved problems for evaluating limits of algebraic functions by using limit formulas for learning and practicing.