Find the limit of the this equation

[shwanky]

Homework Statement

Evaluate the limit if it exists.

Homework Equations

$$lim_{x\to2} \frac{\sqrt{x-6}-2}{\sqrt{x-3}-1}$$

The Attempt at a Solution

Completely lost, I've tried, finding the conjugate of both the numerator or denominator and I'm unable to find a solution.

 

[HallsofIvy]

Are you sure that is the problem? AT x= 2, the fraction is
$$\frac{2i- 2}{i- 1}= 2$$
Nothing hard about that!

Perhaps you meant
[tex]\frac{\sqrt{6- x}- 2}{\sqrt{3- x}-1}[/itex]
which is "0/0" when x= 2.
Use the "conjugates" as you mention- multiply both numerator and denominator by $(\sqrt{x-6}+2)(\sqrt{x-3}+1)$. You will get x²-4 in the numerator and x-2 in the denominator and can cancel.

 

[shwanky]

ok, here's another one I don't understand.

Homework Equations

$$lim_{t \to 0} \frac{x+1}{xsin(\Pi x)}$$

The Attempt at a Solution

$$lim_{t \to 0} \frac{x+1}{xsin(\Pi x)}(\frac{x-1}{x-1})$$
$$lim_{t \to 0} \frac{(x^2-1)}{(x^2-1)sin(\Pi x)}$$
$$lim_{t \to 0} \frac{(1)}{sin(\Pi x)}$$

At this point I get stuck... the $$sin(\Pi x)$$ is still 0. I tried using some of the trig identities, that I actually remembered, and got this.

$$lim_{t \to 0} \frac{(sin^2(\Pi x) + cos^2(\Pi x)}{sin(\Pi x)}$$

Now I thought if I could get I could get the numerator in terms of sine, I could simple divide into it, but I'm not sure if this would work :-/... any suggestions?

 

[shwanky]

is it possible for me to do

$$lim_{t \to 0} \frac{(1)}{sin(\Pi x)}$$
$$lim_{t \to 0} \frac{(1)}{csc(\Pi x)}$$
$$lim_{t \to 0} sin(\Pi x)$$

?

 

[shwanky]

mmm never mind. there was a problem in my solution. $$xsin(\Pi x)(x-1)$$ does not equal $$(x2-1)sin(\Pi x)$$... I will try again...

 

[shwanky]

ok, I think I got it.

$$lim_{t \to 1} \frac{x+1}{xsin(\Pi x)}$$

$$lim_{t \to 1} \frac{x+1}{xsin(\Pi x)}(\frac{csc(\Pi x)}{csc(\Pi x)})$$

$$lim_{t \to 1} \frac{(x+1)(csc(\Pi x)}{x}$$

$$lim_{t \to 1} \frac{(x+1)(csc(\Pi x)}{x}$$

$$lim_{t \to 1} csc(\Pi x) + \frac{(csc(\Pi x)}{x}$$

$$lim_{t \to 1} csc(\Pi * 1) + \frac{(csc(\Pi *1)}{1}$$

$$lim_{t \to 1} csc(\Pi) + (csc(\Pi)$$

$$lim_{t \to 1} 2csc(\Pi)$$

$$lim_{t \to 1+} 2csc(\Pi) = +\infty$$

$$lim_{t \to 1-} 2csc(\Pi) = -\infty$$

 

[shwanky]

Sorry, the limit is t->1+ not t->0...

 

[shwanky]

ok, now if I take the left hand limit and right hand limit, the right hand limit approaches infinity while the left hand appraches negative infinity?

 

[HallsofIvy]

Does it bother you at all that you are taking the limit as t-> 1 but there is no t in the formula?! (Mathematics requires precision- be careful what you write!)

Perhaps you meant
$$lim_{t \to 1^+} \frac{t+1}{tsin(\Pi t)}$$

I don't see difficulty with that. The denominator is continuous and at t= 1 is $(1)sin(\pi)= 0$ but the numerator does NOT go to 0. What does that tell you?

(Don't capitalize "Pi"- you want \pi: $\pi$.)

 

[shwanky]

... I am such a dork...

$$\lim_{t \to 1^+} \frac{t+1}{tsin(\pi t)} = -\infty$$ as $$t \to 1^+$$