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
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 : $\lim_{x->1}\sqrt[3]{x} = \lim_{x->1}(x)^{1/3} = \sqrt[3]{\lim_{x->1}x}$
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
12th grade math
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