There are some fundamental properties in limits. They are used as formulas in some basic operations and also used in evaluating limits of the functions in calculus.

The list of fundamental operations of limits with their formulas and proofs.

$(1).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f(x)+g(x)\Big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ $+$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$

$(2).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f(x)-g(x)\Big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ $-$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$

$(3).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f(x) \times g(x)\Big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ $\times$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$

$(4).\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}}{\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}}$

The limit of reciprocal of a function is equal to the reciprocal of limit of the function.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{1}{f(x)}}$ $\,=\,$ $\dfrac{1}{\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}}$

The limit of product of a constant and a function is equal to the product of constant and the limit of the function.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[k.f{(x)}\Big]}$ $\,=\,$ $k \times \displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$

The limit of an exponential function is equal to exponentiation of their limits.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize {f{(x)}}^{g{(x)}}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}}$

The limit of composition of two functions is equal to the value of the function for the limit of its internal function.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}\Big)$

List of standard results of limits with proofs to use them as formulas in calculus.

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