Evaluate the limit $\lim_{x\to 0} \frac{(\tan(x+\pi/4))^{1/3}-1}{\sin(2x)}$ Evaluate the limit
$$\lim_{x\to 0} \frac{(\tan(x+\pi/4))^{1/3}-1}{\sin(2x)}$$
I know the limit is $1\over3$ by looking at the graph of the function, but how can I algebraically show that that is the limit. using this limit: $$\lim_{x \rightarrow 0} \frac{(1+x)^c -1}{x} =c$$? (without L'Hopital Rule)

Use the definition of derivative:  $$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}.$$  Second hint:  multiply numerator and denominator by $x$.
