Limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left

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In summary, the limit of the expression \frac{\sqrt{x}-2}{x-4} as x approaches 4 from the left is \frac{1}{\sqrt{x}+ 2}. This can be found by rationalizing the numerator or using L'Hopital's rule.
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Homework Statement


[tex]\lim_{x\rightarrow 4^{-}}\frac{\sqrt{x}-2}{x-4}[/tex]
Evaluate

The Attempt at a Solution


Well, I used a calculator to substitute arbitrary values to see what it appears to approach.

input | output
2 .293
3 .268
3.5 .258
3.9 .252

So I concluded that the the limiting value is in fact .25.. and I was wrong. How do I evaluate this?

NOTE THAT WE ARE APPROACHING FROM THE LEFT!
 
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  • #2


Use L'Hospital's rule.
 
  • #3


Thank you, I got it.
 
  • #4


A fairly standard way of handling a problem like that, and simpler than apply L'Hopital's rule is to rationalize the numerator. Multiplying both numerator and denominator by [itex]\sqrt{x}+ 2[/itex] you get
[tex]\frac{(\sqrt{x}-2)(\sqrt{x}+2)}{(x-4)(\sqrt{x}+2)}= \frac{x- 4}{(x-4)(\sqrt{x}+ 2)}[/tex]
which, for x> 4, is
[tex]\frac{1}{\sqrt{x}+ 2}[/tex]
Of course, the limit of that, as x goes to 2, is the same as the limit of the original problem.
 
  • #5


HallsofIvy said:
A fairly standard way of handling a problem like that, and simpler than apply L'Hopital's rule is to rationalize the numerator. Multiplying both numerator and denominator by [itex]\sqrt{x}+ 2[/itex] you get
[tex]\frac{(\sqrt{x}-2)(\sqrt{x}+2)}{(x-4)(\sqrt{x}+2)}= \frac{x- 4}{(x-4)(\sqrt{x}+ 2)}[/tex]
which, for x> 4, is
[tex]\frac{1}{\sqrt{x}+ 2}[/tex]
Of course, the limit of that, as x goes to 2, is the same as the limit of the original problem.

Yeah, that is what I did. I'm not up to the rule yet. Thanks again halls of ivy.
 

1. What is the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left?

The limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left is -4.

2. How do you calculate the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left?

To calculate the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left, we can use the direct substitution method. This means plugging in 4 for x and solving the expression to find the limit.

3. Is the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left equal to 4?

No, the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left is not equal to 4. As mentioned before, the limit is actually equal to -4.

4. Why is the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left equal to -4?

This is because when x approaches 4 from the left, the expression (sqrt(x) - 2) becomes a negative value, as the square root of 4 is 2 and subtracting 2 gives us 0. This negative value is then multiplied by the value of (x-4), which is also negative as x is approaching 4 from the left. Therefore, the overall result is a positive value of -4.

5. Can the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left be changed by manipulating the expression?

No, the limit of (sqrt(x) - 2)(x-4) as x approaches 4 from the left is a constant value of -4 and cannot be changed by manipulating the expression. The limit is determined by the behavior of the expression as x approaches 4 from the left, and this behavior cannot be altered by changing the expression itself.

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