Question

More limits..

Answer 1

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

Answer 2

Yeah that would be my first choice too, to multiply by this form of 1 and see what happens:

\[\large \frac{\sqrt{x^2+9}+5}{\sqrt{x^2+9}+5}\]

Answer 3

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

Answer 4

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

Answer 5

hint: First define what type of form it falls under

Answer 6

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

Answer 7

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.

Answer 8

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

Answer 9

ah k @Empty let me try hahaa

Answer 10

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. \[\lim_{x\rightarrow -4} \frac{\frac{d}{dx}(\sqrt{x^2+9}-5)}{\frac{d}{dx}(x+4)}\]\[\lim_{x\rightarrow -4}\frac{\frac{1}{2}(x^2+9)^{-1/2} \cdot 2x}{1}\]\[\lim_{x\rightarrow -4} \frac{\dfrac{x}{(x^2+9)^{1/2}}}{1}\]

Answer 11

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

Answer 12

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

Answer 13

I'm having trouble with the denominator :|

Answer 14

@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

Answer 15

Yeah i just realized that, sorry about that

Answer 16

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!

Answer 17

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

Answer 18

Awesome. Notice anything about the denominator now? :D

Answer 19

(Sorry, went to have lunch)

Answer 20

It has an x+4 in it!

Answer 21

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

Answer 22

(with a +5 in the denominator, sorry)

Answer 23

Awesome, now try taking the limit. :P

Answer 24

em it says error o.o

Answer 25

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

Answer 26

@Empty

Answer 27

do it :U

Answer 28

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

Answer 29

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

Answer 30

oh. -8/10

Answer 31

yay c:

Answer 32

drum roll!!!

Answer 33

hahah