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# Is the function $f(x)=\Large\frac{3x+4\tan x}{x}$ continuous at x = 0? If not, how may the function be defined to make it continuous at this point.

Toolbox:
• For continuity $LHL = RHL = f(0)$
• $\lim\limits_{x \to 0}\large\frac{tan\:x}{x}=1$
LHL = RHL
$\lim\limits_{x \to 0}\frac{3x+4tanx}{x}=7$
$\lim\limits_{x \to 0}\frac{3x}{x}+ \lim\limits_{x \to 0}\frac{4tanx}{x}$
=3+4=7

$f(0)=7$ so as to make $f$ a continuous function.
$f(x)= \left\{ \begin{array}{1 1} \Large\frac{3x+4tanx}{x} & \quad ,\;x\neq0 \\ 7 & \quad ,\;x=0 \end{array} \right.$

edited Mar 26, 2013