Limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4 - Simple Limits Question

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In summary, to evaluate the limit as x approaches 4 of (2 - √x)/(3 - √(2x + 1)), you need to first rationalize the numerator and denominator by multiplying by (2 + √x)/(2 + √x) and (3 + √(2x + 1))/(3 + √(2x + 1)). This will simplify the expression and allow you to take the limit, which will result in a divide by zero error.
  • #1
Yalladoonia
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Homework Statement



Evaluate the following limit
[itex]
\displaystyle\lim_{x\rightarrow 4} {\frac{2-\sqrt{x}}{3-\sqrt{2x+1}}}[/itex]

Homework Equations



lim x -> 4

The Attempt at a Solution


i tried change of variables so i get [itex]√2x+1 = u[/itex]
then i rearranged that to get [itex]u^2-1/2[/itex]
And i rearrnage original equation and i get this..
[itex]-(u^2-7)/3-u(u^2-1/2)[/itex]

Sorry i am new with this latex thing .. i don treally konw how to use it the sqrt thing so i just lay it out this way
 
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  • #2
I am really unsure of how to use this latex thing... but personally i think my attempt at the solution is all garbage so if anyone could guide me on the right path that would be help ful
 
  • #3
Multiply with the nominator and denominator with 3+sqrt(4x+1)
 
  • #4
Or l'hopital's?
dirk_mec1 said:
Multiply with the nominator and denominator with 3+sqrt(4x+1)
##\frac{2 - \sqrt{x}}{3 - \sqrt{2x + 1}}\frac{3+\sqrt{2x+1}}{3+\sqrt{2x+1}}=\frac{(2-\sqrt{x})(3+\sqrt{2x+1})}{9 - (2x +1)}##
which still has a divide by zero error as x->4
 
  • #5
i think l'hopitals is derivatives, i haven't learned that yet I am only allowed to use limits
 
  • #6
You need to rationalize both the numerator and denominator. To do this, multiply by the following:
$$ \frac{2 + \sqrt{x}}{2 + \sqrt{x}} \cdot \frac{3 + \sqrt{2x + 1}}{3 + \sqrt{2x + 1}}$$

When you do this, you get something that you can simplify and then take the limit.
 

FAQ: Limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4 - Simple Limits Question

1. What is the limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4?

The limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4 is undefined. This is because when x approaches 4, the denominator of the fraction approaches 0, which is undefined in mathematics.

2. Can the limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4 be evaluated using direct substitution?

No, direct substitution cannot be used to evaluate the limit in this case. This is because direct substitution can only be used when the limit is a finite number, but in this case, the limit is undefined.

3. How can the limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4 be evaluated?

The limit can be evaluated using algebraic manipulation and the concept of conjugate pairs. By multiplying the numerator and denominator by the conjugate of the denominator, the expression can be simplified and the limit can be evaluated as x approaches 4.

4. Is the limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4 equal to the limit of (2+sqrt(x))/(3+sqrt(2x+1)) as x approaches 4?

Yes, the limit of (2-sqrt(x))/(3-sqrt(2x+1)) as x approaches 4 is equal to the limit of (2+sqrt(x))/(3+sqrt(2x+1)) as x approaches 4. This is because the two expressions are equivalent and both result in the undefined limit as x approaches 4.

5. What is the significance of finding the limit of a function?

Finding the limit of a function is important in understanding the behavior of the function as the input value gets closer and closer to a specific value. It can also help determine if the function is continuous at that value. In some cases, the limit may not exist, indicating a discontinuity in the function.

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