Limits involving L'hospital's rule question?

Question

I am stuck on this one problem:

Lim as x->0+ ((3x+1)sinx - x)/(xsinx)

The way I worked it, I got 2, but
I think it maybe wrong. Any help would be great.

PS: The problem I seem to be having is that I don't know what the derivative of 3sinx would be. I don't know if it is just cosx or 3cosx. I do not know if I need to drop the 3 or not when I take the derivative a second time.

Another quick unrelated question. Does 10/3 work as the only critical number for the function f(x) = x*(5-x)^.5

Answer 1

differentiating
(3x+1)cosx+sinx(3)-1/xcosx+sinx
applyingthe limits
=4-1/0
differentiating againn
(3x+1)(-sinx)+cosx(3)+3cosx/-xsinx+cosx+cosx
applying the limits
3+3/2
3

Answer 2

First you have to check that the numerator and denominator are both 0 for x=0. They are.
Then you can take the derivative of both numerator and denominator:
Lim as x->0+ ((3x+1)cos(x) + 3 sin(x) - 1) / (x cos(x) + sin(x))

Using this new expression, both numerator and denominator are still 0. So, do the process again:
Lim as x->0+ (-(3x+1) + 3 cos(x) + 3 cos(x)) / (-x sin(x) + cos(x) + cos(x))

Then evaluate that at x=0. This time, the numerator equals 6, and the denominator equals 2.
6 / 2 = 3.

3 is the answer.

Answer 3

The answer is 3. Differentiating the top and the bottom twice gives:
Lim{x->0} of 3+1/(x - 2 Cot x). The right term goes to zero as x->0, so the limit is 3.

Answer 4

lim (3^t - 2^t)/t t -> 0 first of all, to substantiate that we've the style [0/0], we plug in t = 0 for the numerator to confirm if we get 0. 3^t - 2^t = 3^0 - 2^0 = a million - a million = 0 and the denominator of direction is 0, so we get [0/0], subsequently permitting us to be conscious L'wellbeing facility's rule. Doing so, we notice that the spinoff of three^t is 3^t ln(3), and the spinoff of two^t is two^t ln(2). Differentiating precise and backside, we get lim (3^t ln(3) - 2^t ln(2) ) / ( a million ) t -> 0 lim 3^t ln(3) - 2^t ln(2) t -> 0 Plug in t = 0, to get 3^0 ln(3) - 2^0 ln(2) or ln(3) - ln(2)