Question

Evaluate: lim x-> 0 sinx/(x+tanx)

Answer 1

@SithsAndGiggles

Answer 2

Quick question, have you learned L'Hopital's rule? It's not required, but it's one way to go about this.

Answer 3

Nope, not yet. (only ch 3 right now, just starting to take derivatives)

Answer 4

Another way would be to make use of the fact that near values of \(x=0\), you have \(\sin x\approx\tan x\approx x\). This is the simplest way.

Answer 5

Yet another way is to use identities and known limits like before. Your choice.

Answer 6

Can you show me the way with identities and limits since I'm assuming that's how the textbook wants?

Answer 7

Sure.
\[\large\begin{align*}
\lim_{x\to0}\frac{\sin x}{x+\tan x}&=\lim_{x\to0}\frac{\sin x}{x+\frac{\sin x}{\cos x}}\\\\
&=\lim_{x\to0}\frac{1}{\frac{x}{\sin x}+\frac{1}{\cos x}}&\text{divide num and den by }\sin x\\\\
&=\dfrac{\displaystyle\lim_{x\to0}1}{\displaystyle\lim_{x\to0}\frac{x}{\sin x}+\lim_{x\to0}\frac{1}{\cos x}}
\end{align*}\]

Answer 8

*divide num and den by \(\sin x\)*

Answer 9

Ok got it! Thanks!