Solve Limit x→2: (4-(√18-x))/(x-2)

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The limit as x approaches 2 for the expression (4 - √(18 - x)) / (x - 2) results in an indeterminate form, prompting the need for simplification. The discussion highlights the use of the conjugate to simplify the numerator, although initial attempts still yield an indeterminate result upon substitution. Participants suggest that careful distribution and simplification can lead to cancellation, and one contributor notes that L'Hôpital's rule may not be effective for this problem. Ultimately, the discussion reveals confusion about the limit's existence, with one participant indicating that their graph suggests no limit as it approaches positive and negative infinity. The conversation emphasizes the importance of proper algebraic manipulation to understand the behavior of the function near the limit.
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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?
 
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Have you learned about L'Hoptial's rule yet?
 
Ritzycat said:

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.
 
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.
 
Ritzycat said:
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.
 
B3NR4Y said:
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.
 
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.
 
Ritzycat said:
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.
 
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.
 
  • #10
Multiply the numerator and denominator by the conjugate of ##4 - \sqrt{18-x}##. From there, it should work out if you simplify everything right.
 
  • #11
AMenendez said:
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.
 
  • #12
Ritzycat said:
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/
 

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