More limits..

- Babynini

More limits..

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- chestercat

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- Babynini

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- Babynini

I'm trying to rationalize but I keep going off somewhere.

- Empty

Yeah that would be my first choice too, to multiply by this form of 1 and see what happens:
\[\large \frac{\sqrt{x^2+9}+5}{\sqrt{x^2+9}+5}\]

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## More answers

- anonymous

\[\lim_{x \rightarrow -4}\frac{ \sqrt{x^2+9}-5 }{ x+4 }\]

- Babynini

@Empty the problem is i'm not quite sure how to multiply those..heh

- Jhannybean

hint: First define what type of form it falls under

- Jhannybean

Just judging by the left and right hand limit type, and the form, you can apply LH rule to this.

- Empty

In a lot of ways I really look at calculus as 'mastering algebra'
Let's just look at the numerator or denominator one at at time, first the numerator:
\[(\sqrt{x^2+9}-5)(\sqrt{x^2+9}+5)\]
What do we do? Well, it turns out we distribute, just like we would here even though it looks more complicated, try to fit it to this pattern if it helps you keep things straight:
\[(a-b)(a+b)=a*a+a*b-b*a-b*b = a^2-b^2\] (notice the middle terms cancel out, which is super handy to save time)
Give it a shot, the work you put in now will make a difference later in the semester and during tests.

- anonymous

\[\left( \sqrt{x^2+9}-5 \right) \times \frac{ \left( \sqrt{x^2+9}+5 \right) }{ \left( \sqrt{x^2+9}+5 \right) }=\frac{ \left( \sqrt{x^2+9} \right)^2-5^2}{\sqrt{x^2+9}+5}\]

- Babynini

ah k @Empty let me try hahaa

- Jhannybean

So you want to simply this function to a form where you can input your limit value to break free from the \(\dfrac{0}{0}\) form.
Take the derivative of the numerator an the denominator. \[\lim_{x\rightarrow -4} \frac{\frac{d}{dx}(\sqrt{x^2+9}-5)}{\frac{d}{dx}(x+4)}\]\[\lim_{x\rightarrow -4}\frac{\frac{1}{2}(x^2+9)^{-1/2} \cdot 2x}{1}\]\[\lim_{x\rightarrow -4} \frac{\dfrac{x}{(x^2+9)^{1/2}}}{1}\]

- Babynini

so for the numerator, does it equal:
x^2 + 9 - 25?

- Jhannybean

\[\lim_{x\rightarrow -4} \frac{\dfrac{x}{(x^2+9)^{1/2}}}{1} =\frac{-4}{((-4)^2+9)^{1/2}} = \color{red}{-\frac{4}{5}} \]

- Babynini

I'm having trouble with the denominator :|

- Babynini

@Jhannybean I'm not really sure what all you did because I have no idea how to do derivatives or anything..
so I need to do it algebraically o.o

- Jhannybean

Yeah i just realized that, sorry about that

- Empty

Yes you're correct with the numerator, however there are two simplifications you can do! combine the 9 and -25 and then you can factor this as the difference of two squares. Try that out, and I'll help you worry about the bottom since you don't want to distribute the bottom actually!

- Babynini

Okay!
the top then is (x-4)(x+4)

- Empty

Awesome. Notice anything about the denominator now? :D

- Babynini

(Sorry, went to have lunch)

- Babynini

It has an x+4 in it!

- Babynini

\[\frac{ x-4 }{ \sqrt{x^2+9} }\]

- Babynini

(with a +5 in the denominator, sorry)

- Empty

Awesome, now try taking the limit. :P

- Babynini

em it says error o.o

- Babynini

(-4-4)/ [sq{-4^2+9)}+5]

- Babynini

@Empty

- zepdrix

do it :U

- Babynini

It's not working! because it ends up being a negative under the sq root..

- zepdrix

\(\large\rm -4^2\ne(-4)^2\)
careful how you square.

- Babynini

oh. -8/10

- zepdrix

yay c:

- IrishBoy123

drum roll!!!

- Babynini

hahah

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