Limits problem

1. Sep 25, 2008

ganondorf29

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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. Im not too sure what Im supposed to do now

2. Sep 25, 2008

sutupidmath

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

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

3. Sep 25, 2008

ganondorf29

No, we cant use l'hopitals rule because we haven't learned it yet.

4. Sep 25, 2008

sutupidmath

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!

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