Evaluate the limit as y approaches 4

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In summary, the problem asks to evaluate the limit as y approaches 4 of the function (y-3√y+2)/(√y-2). The numerator can be factored as (√y-2)(√y-1), making the limit as y approaches 4 equal to 1. The function is undefined at y=4, but the limit only considers values close to 4, not at 4 itself. The square roots in the function can also be rewritten in terms of u to make factoring easier.
  • #1
hunt3rshadow
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Homework Statement



Evaluate lim y→4

y-3√y + 2
√y - 2



The Attempt at a Solution



If I plug 4 into Y, my answer is undefined. But if I do the chart method, where I plug in 3.999 and 4.0001 the answer is 1. So I'm not quite sure what I'm doing wrong.
 
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  • #2
Try factoring the numerator! The square roots might throw you off a bit, but it's in the same form of a quadratic equation.

You can't plug in 4 directly because the function isn't defined at 4. But limits don't concern themselves with what happens at that point, only what happens very close to that point.
 
  • #3
hunt3rshadow said:

Homework Statement



Evaluate lim y→4

y-3√y + 2
√y - 2



The Attempt at a Solution



If I plug 4 into Y, my answer is undefined. But if I do the chart method, where I plug in 3.999 and 4.0001 the answer is 1. So I'm not quite sure what I'm doing wrong.

The numerator is quadratic in form, and can be factored.

Or, you can let u = √y, and rewrite the fraction in terms of u, and factor the numerator. With this change, the limit will be as u → 2.
 
  • #4
Mark and Scurty, thank you. Mark I get your second method, but for both of you, I don't understand how to factor the numerator. Is it really factor-able? I don't get what value to factor by.
 
Last edited:
  • #5
hunt3rshadow said:

Homework Statement



Evaluate lim y→4

y-3√y + 2
√y - 2

The Attempt at a Solution



If I plug 4 into Y, my answer is undefined. But if I do the chart method, where I plug in 3.999 and 4.0001 the answer is 1. So I'm not quite sure what I'm doing wrong.
Of course 4.001 & 3.999 won't give you exactly 1. They give 0.99975 and 1.00025 respectively.
 
  • #6
Can you use parenthesis? I can't tell what your sqrt is encompassing.
 
  • #7
USN2ENG said:
Can you use parenthesis? I can't tell what your sqrt is encompassing.

Just the Y. Nothing else.
 
  • #8
Parentheses aren't needed. √y is just the square root of y.
 
  • #9
Can someone tell me how exactly do I factor the numerator?
 
  • #11
Or if you use my suggestion, you're factoring u2 - 3u + 2.
 
  • #12
Mark44 said:
(√y - ?)(√y - ?)

Like that...

I totally forgot factoring square roots. Thanks. Saved me alotta trouble.
 
Last edited:

FAQ: Evaluate the limit as y approaches 4

What does it mean to evaluate the limit as y approaches 4?

Evaluating the limit as y approaches 4 means finding the value that a function approaches as the input variable, y, gets closer and closer to the value of 4. This is also known as the limiting value or the limit at 4.

Why is it important to evaluate limits?

Evaluating limits is important because it allows us to understand the behavior of a function near a specific point. It can also help us determine whether a function is continuous at that point, and can be used to solve more complex problems in calculus and other areas of mathematics.

What is the process for evaluating a limit as y approaches 4?

The process for evaluating a limit as y approaches 4 involves plugging in values of y that are very close to 4 on either side of the point and observing the resulting output values. If the output values are approaching a specific value, that is the limit. If the output values are approaching different values from each side, the limit does not exist.

Can a limit have more than one value?

No, a limit can only have one value. If the output values are approaching different values from each side, the limit does not exist. This is known as a discontinuity at that point.

What are some real-world applications of evaluating limits?

Evaluating limits has many real-world applications, such as determining the maximum safe speed for a car on a curved road, finding the maximum and minimum values of a function, and calculating the rate of change in a physical process. It is also used in fields such as economics, physics, and engineering to model and analyze various systems and phenomena.

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