Find the constant such that f(x)= {-4*((sin x)/(x)), x=0 is continuous on the entire real line. a)4 b)-4 c)-10 d)10 e)1 **I know that the answer is -4(B), but I need to know how to get it**

Question

Find the constant such that

f(x)= {-4*((sin x)/(x)), x<0
-------{a + 10x, x>=0

is continuous on the entire real line.
a)4
b)-4
c)-10
d)10
e)1
**I know that the answer is -4(B), but I need to know how to get it**

anonymous · Answer

Help please, I have test corrections

anonymous · Answer

Use this to take a left sided limit.
$$\lim_{x \rightarrow 0}\frac{ \sin x }{ x }=1$$
Then take a limit of the right side approaching 0 of $a+10x$.
The limits have to be equal to each other and f(0) for f(x) to be continuous.

anonymous · Answer

@peachpi how is sin x/x = 1? that was what stumped me

anonymous · Answer

(sin x)/x isn't 1. The limit approaching 0 is.
$$\lim_{x \rightarrow 0}\frac{ \sin x }{ x }=1$$
and 
$$\lim_{x \rightarrow 0}\frac{ 1-\cos x }{ x }=0$$
are two limits you need to be able to recognize on sight.
I'll see if I can find links to proofs.

anonymous · Answer

For the sine limit: http://math.ucsb.edu/~jcs/SqueezeTheorem.pdf For the cosine limit: http://math.hws.edu/~mitchell/Math130F12/tufte-latex/TrigLimits.pdf