# How do you use the definition of continuity and the properties of limits to show that the function f(x) = 5x^4 - 9x^3 + x - 7 is continuous at a given number a=7?

Feb 25, 2016

${\lim}_{x \rightarrow 7} f \left(x\right) = {\lim}_{x \rightarrow 7} \left(5 {x}^{4} - 9 {x}^{3} + x - 7\right)$

$= {\lim}_{x \rightarrow 7} \left(5 {x}^{4}\right) - {\lim}_{x \rightarrow 7} \left(9 {x}^{3}\right) + {\lim}_{x \rightarrow 7} \left(x\right) - {\lim}_{x \rightarrow 7} \left(7\right)$

(Limit of a sum and difference)

$= 5 {\lim}_{x \rightarrow 7} \left({x}^{4}\right) - 9 {\lim}_{x \rightarrow 7} \left({x}^{3}\right) + {\lim}_{x \rightarrow 7} \left(x\right) - {\lim}_{x \rightarrow 7} \left(7\right)$

(Limit of a constant multiple of a function)

$= 5 {\left({\lim}_{x \rightarrow 7} x\right)}^{4} - 9 {\left({\lim}_{x \rightarrow 7} x\right)}^{3} + {\lim}_{x \rightarrow 7} \left(x\right) - {\lim}_{x \rightarrow 7} \left(7\right)$

(Limit of a power or repeated limits of products.)

$= 5 {\left(7\right)}^{4} - 9 {\left(7\right)}^{3} + \left(7\right) - \left(7\right)$

(Limits of identity and constant functions)

$= f \left(7\right)$

We have shown that ${\lim}_{x \rightarrow 7} f \left(x\right) = f \left(7\right)$, so, by the definition of continuity (at a point), $f$ is continuous at $7$