# Properties Of Limits

Properties of Limits : Let 'b' and 'c' are any real numbers. Let 'n' be any positive integer and let 'f' and 'g' be functions with the limits.
$\lim_{x->c}f(x)$ = L and $\lim_{x->c}g(x)$ = K

1) Scalar multiple :
$\lim_{x->c}bf(x)$ = bL

If there is any constant real number outside the limit then we can take that number outside the limit.
Example : $\lim_{x->0}4 cos(x)$ = 4 $\lim_{x->0} cos(x)$

2) Sum of two limits :
$\lim_{x->c}f(x) + g(x)$ = $\lim_{x->c}f(x)$ + $\lim_{x->c}g(x)$ = L + K

Example : $\lim_{x->2}(4x^{2} + 3)$ = $\lim_{x->2}(4x^{2})$ + $\lim_{x->2}(3)$

This property is not limited to only two functions.
3) Difference of two limits :
$\lim_{x->c}f(x) - g(x)$ = $\lim_{x->c}f(x)$ - $\lim_{x->c}g(x)$ = L - K

Example : $\lim_{x->2}(4x - 3)$ = $\lim_{x->2}(4x)$ - $\lim_{x->2}(3)$

This property is not limited to only two functions.
4) Product rule for limit :
$\lim_{x->c}f(x)g(x)$ = $\lim_{x->c}f(x)$ * $\lim_{x->c}g(x)$ = L * K

Example : $\lim_{x->0}x*(cos(x)$ = $\lim_{x->0}(x)$ * $\lim_{x->0}cos(x)$

This property is not limited to only two functions.

## Properties of Limits

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5) Quotient rule :

$\lim_{x->c}\left [\frac{f(x)}{g(x)}\right]$ = $\frac{\lim_{x->c}f(x)}{\lim_{x->c}g(x)}$ = $\frac{L}{K}$

provided $\lim_{x->c}g(x) \neq$0 , K $\neq$0

Example : $\lim_{x->2}\left [\frac{x^{2}}{x-1}\right]$ = $\frac{\lim_{x->2}(x^{2})}{\lim_{x->2}{}(x-1)}$

6) Power rule :
$\lim_{x->c}[f(x)]^{n}$ = = $L^{n}$

Example : $\lim_{x->1}\sqrt{x} = \lim_{x->1}(x)^{1/3} = \sqrt{\lim_{x->1}x}$

7) $\lim_{x->a}c$ = c

Example : $\lim_{x->2}3$ = 3

8) $\lim_{x->a}x$ = a

Example : $\lim_{x->5}x$ = 5

9) $\lim_{x->a}x^{n}$ = $a^{n}$

Example : $\lim_{x->2}x^{3}$ = $2^{3}$ = 8