# Properties of Limits

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.

### Operations

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

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

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

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

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

### Constant multiple

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

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

### Reciprocal rule

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

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

### Exponentiation

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

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

### Composition

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

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

### Formulas

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

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