Solve Limit x→2: (4-(√18-x))/(x-2)

[Ritzycat]

Homework Statement

[/B]
lim x→2
(4-(√18-x)) / (x-2)

Note that the square root goes over the 18 AND the x, not just the 18.

I don't know how to use the fancy mathematical notation on this forum and I have no idea where to go to find out how to use it.

Homework Equations

There are absolutely NO relevant equations. ZERO!

The Attempt at a Solution

I multiplied the numerator and the denominator by the conjugate of the numerator, because direct substitution would yield an indeterminate. However, after distributing, direct substitution of 2 still gave me an indeterminate.

Here is my distribution
(16-(18-x)) / (4x-x(√18-x)-8+(√18-x))

How can I further simplify this?

 

[B3NR4Y]

Have you learned about L'Hoptial's rule yet?

 

[gopher_p]

Homework Statement

[/B]
lim x→2
(4-(√18-x)) / (x-2)

Note that the square root goes over the 18 AND the x, not just the 18.

I don't know how to use the fancy mathematical notation on this forum and I have no idea where to go to find out how to use it.

Homework Equations

There are absolutely NO relevant equations. ZERO!

The Attempt at a Solution

I multiplied the numerator and the denominator by the conjugate of the numerator, because direct substitution would yield an indeterminate. However, after distributing, direct substitution of 2 still gave me an indeterminate.

Here is my distribution
(16-(18-x)) / (4x-x(√18-x)-8+(√18-x))

How can I further simplify this?

Don't FOIL the denominator. Simplify the numerator. You should see a fairly easy cancellation.

 

[Ritzycat]

Have you learned about L'Hoptial's rule yet?

No, I haven't!

Don't FOIL the denominator. Simplify the numerator. You should see a fairly easy cancellation.

I think I see what you mean, but doesn't 16-(18-x) simplify to (-2 - x)? My denominator has (x-2) in it.

 

[gopher_p]

I think I see what you mean, but doesn't 16-(18-x) simplify to (-2 - x)? My denominator has (x-2) in it.

You didn't distribute the minus sign.

 

[Mark44]

Have you learned about L'Hoptial's rule yet?

For problems like this, L'Hopital's rule is often ineffective. The approach taken by the OP is the better way to go here.

 

[Ritzycat]

DUH. Can't believe I forgot to do that. Makes much more sense now.

However, after I simplify and cancel, I'm left with a denominator that = 0 when x = 2.

-1 / (4 - (√18 - x))

After I graphed it, I found there's no limit. (Approaching positive and negative infinity) However, how do I algebraically prove that there is no limit in a function? Sorry for all of these questions - I'm studying limits on my own so I might be missing certain important concepts here and there.

 

[Mark44]

DUH. Can't believe I forgot to do that. Makes much more sense now.

However, after I simplify and cancel, I'm left with a denominator that = 0 when x = 2.

-1 / (4 - (√18 - x))

After I graphed it, I found there's no limit. (Approaching positive and negative infinity) However, how do I algebraically prove that there is no limit in a function? Sorry for all of these questions - I'm studying limits on my own so I might be missing certain important concepts here and there.

I think you made a mistake. You should end up with 4 + √(18 - x) in the denominator.

Note the parentheses I used to indicate that what's under the radical is 18 - x, not just 18.

 

[MayCaesar]

I would start with making a replacement $y=x-2$. You can further simplify this by taking $z=\frac{y}{16}$. After that just follow the way you did before, but the expression should look much more pleasant now.

 

[AMenendez]

Multiply the numerator and denominator by the conjugate of $4 - \sqrt{18-x}$. From there, it should work out if you simplify everything right.

 

[AMenendez]

Multiply the numerator and denominator by the conjugate of $4 - \sqrt{18-x}$. From there, it should work out if you simplify everything right.

Never mind, I didn't see that you had already tried that.

 

[Fredrik]

I don't know how to use the fancy mathematical notation on this forum and I have no idea where to go to find out how to use it.

Click "info", "help/how-to" and "latex primer"...or use this link: https://www.physicsforums.com/help/latexhelp/