Question

solve: lim(x->0) of (1 - cosx + 3sinx)/x

Answer 1

\[
\lim_{x\to 0}\frac{1-\cos x+3\sin x}{x} = -\lim_{x\to 0}\frac{\cos x-1}{x}+3\lim_{x\to 0}\frac{\sin x}{x}
\]

Answer 2

These two limits are worth remembering.

Answer 3

ok

Answer 4

what is the next step

Answer 5

Look up those limits

Answer 6

the answer is 3

Answer 7

other than knowing those two limits, are there any other ways to solve (aside from finding those limits by other means)

Answer 8

wio has the most simplest way
but you could use l'hospital
since 1-cos(0)+3sin(0)=1-1+0=0 and 0=0
we have the case 0/0
So we can apply the l'hospital:
\[\lim_{x \rightarrow 0}\frac{1-\cos(x)+3\sin(x)}{x}=\lim_{x \rightarrow 0}\frac{0+\sin(x)+3\cos(x)}{1}= \lim_{x \rightarrow 0}(\sin(x)+3\cos(x)) \\ =\sin(0)+3\cos(0) =0+3(1)=0+3=3\]