Question

need help with calculus limit problem

Answer 1

\[\lim_{h \rightarrow 0} (\tan3(x+h)-\tan(3x))/h\]

Answer 2

\[\lim_{h \rightarrow 0} \frac{\tan3(x+h)-\tan(3x)}{h}\]
Recall:
\[\tan (A+B) = \frac{\tan A + \tan B}{1-\tan A\tan B}\]
So,
\[\lim_{h \rightarrow 0} \frac{\tan3(x+h)-\tan(3x)}{h}\]\[=\lim_{h \rightarrow 0} \frac{\tan(3x+3h)-\tan(3x)}{h}\]\[=\large\lim_{h \rightarrow 0} \frac{\frac{\tan(3x)+\tan(3h)}{1-\tan3x\tan3h}-\tan(3x)}{h}\]\[=\lim_{h \rightarrow 0} \frac{\tan(3x)+\tan(3h)-(1-\tan3x\tan3h)\tan(3x)}{h(1-\tan3x\tan3h)}\]\[=\lim_{h \rightarrow 0} \frac{\tan(3h)+\tan^2(3x)\tan(3h)}{h(1-\tan3x\tan3h)}\]\[=\lim_{h \rightarrow 0} \frac{\tan(3h)(1+\tan^2(3x))}{h(1-\tan3x\tan3h)}\]\[=\lim_{h \rightarrow 0} \frac{\tan(3h)}{h}\times\lim_{h\rightarrow 0}\frac{(1+\tan^2(3x))}{(1-\tan3x\tan3h)}\]\[=3\lim_{h \rightarrow 0} \frac{\tan(3h)}{3h}\times\lim_{h\rightarrow 0}\frac{(1+\tan^2(3x))}{(1-\tan3x\tan3h)}\]
I think you can proceed from here.

Answer 3

Thanks!!!

Answer 4

@RolyPoly i really appreciate ur help on tht question but please could you help me with another limit problem???

Answer 5

Make a new post :)

Answer 6

thankyou :)