Question

find the values for a and b such that lim x->0 (((sqrt a+bx)-5)/x)=4

Answer 1

Is this right?\[
\lim_{x\to 0}\frac{\sqrt{a+bx}-5}{x}=4
\]

Answer 2

Try multiplying by the conjugate.

Answer 3

I've done that and got:
a+bx-25/x((sqrt a+bx)+5x)=4

Answer 4

You get\[
\lim_{x\to 0}\frac{a+bx-25}{x(\sqrt{a+bx}+5)}
\]

Answer 5

Yes

Answer 6

I'm not sure of the next step

Answer 7

You want the \(x\)s to cancel out. So you want \(a-25=0\)

Answer 8

How would you get the bx's to cancel in order to get a-25

Answer 9

Okay so you want\[
a+bx-25 = bx
\]

Answer 10

Does that make sense?

Answer 11

That way the numerator is divisible by \(x\).

Answer 12

What did you do to get to that step from a+bx-25/(x(sqrt.a+bx)+5)=4

Answer 13

What cancels to get a+bx-25=bx

Answer 14

Okay so look at the denominator: \[
x(\sqrt{a+bx}+5)
\]We doon't want it ot be \(0\).

Answer 15

Exactly, what can be done about the denominator

Answer 16

So if the \(x\) cancel, we have: \[
\lim_{x\to0}\frac{b}{\sqrt{a+bx}+5}
\]

Answer 17

Wouldn't that still yield 0 because an x is on the bottom?

Answer 18

As \(x\to0\) we have \[
\frac{b}{\sqrt{a+b(0)}+5} = \frac{b}{\sqrt{a}+5}
\]

Answer 19

Where did the a go on the numerator as the x's cancelled

Answer 20

Okay, I need to breath in and breath out and summon my patience here.

Answer 21

\(\color{blue}{\text{Originally Posted by}}\) @wio
Okay so you want\[
a+bx-25 = bx
\]
\(\color{blue}{\text{End of Quote}}\)
\(\color{blue}{\text{Originally Posted by}}\) @wio
You want the \(x\)s to cancel out. So you want \(a-25=0\)
\(\color{blue}{\text{End of Quote}}\)

Answer 22

When you get b/((sqrt a)+5)=4 you'll have to solve for "b" by multiplying by ((sqrt a)+5)) on both sides to get b=((sqrt a)+5)+4. After simplification how do you solve for b as b=(sqrt a)+9?

Answer 23

We have two equations: \[
a-25=0\\
b=4(\sqrt{a} + 5)
\]

Answer 24

Sorry it's not +4 it's x4

Answer 25

So a=25 and b=40?

Answer 26

Yes.

Now test it.

Answer 27

Thank you! I understand the process, but how did you know to look for a+bx-25=bx?

Answer 28

Because you want an \(x\) in the numerator and an \(x\) in the denominator to cancel.

Answer 29

So the numerator has to be divisible by \(x\).

Answer 30

So it has to be \(bx\)

Answer 31

Okay, thank you wio!