Limits Problem Homework: Solving for x in (x^2-5x+4) / (sin(x^1/2) - 2)

[ganondorf29]

Homework Statement

lim (x^2-5x+4) / ((sin(x^1/2) - 2))
x->4

Homework Equations

The Attempt at a Solution

Well I factored the top out to be (x-1)(x-4) and if I plug in 4 I get 0 in the numerator. In the denominator, if I plug in 4, I also end up with zero. I am not too sure what I am supposed to do now

 

[sutupidmath]

$$\lim_{x \rightarrow 4}\frac{x^2-5x+4}{sin(\sqrt x-2)}$$

Are u allowed to apply l'hospital's rule? If so, then it will work nicely!

 

[ganondorf29]

No, we can't use l'hospital's rule because we haven't learned it yet.

 

[sutupidmath]

No, we can't use l'hospital's rule because we haven't learned it yet.

Ok, then here it is what just popped into my head,

i would let $$\sqrt x-2=t$$ so when x-->4, t-->0

now

$$\lim_{x \rightarrow 4}\frac{x^2-5x+4}{sin(\sqrt x-2)}=\lim_{x \rightarrow 4}\frac{(x-1)(\sqrt x-2)(\sqrt x+2)}{sin(\sqrt x-2)}=\lim_{t\rightarrow 0}\frac{t}{sin(t)}(t+4)[(t+2)^2-1]$$

now:

$$\lim_{t\rightarrow 0}\frac{t}{sint}=\lim_{t\rightarrow 0}{\frac{1}{\frac{sint}{t}}=1$$

I think the rest is easy!