What is the limit (as u--->2) of square root(4u+1)-3/u-2.

I got 1 as an answer, but I want to check if that's right. I heard it could be something else so I'm unsure and doubtful of my answer.

In most introductory Calculus courses, the study of limits precedes the concept of the derivative, since limits are used to develop the derivative by First Principles, so ...

using your idea of conjugates, .....

Lim ( √(4u+1) - 3)/(u-2) , u--->2
= Lim ( √(4u+1) - 3)/(u-2) * (√(4u+1) + 3))/ (√(4u+1) + 3)) , u--->2
= Lim ( 4u+1 - 9)/ ((√(4u+1) + 3))(u-2))
= lim 4(u-2)/ (√(4u+1) + 3))(x-2)
= lim 4/ (√(4u+1) + 3)) , as u--->2
= 4/6 = 2/3

take derivatives, so the limit is

2/√(4u+1) / 1 = 2/3

I did it by multiplying by the conjugate. I'm not sure of how to do it any other way. Can you explain how that works?

check your text for L'Hospital's (or L'Hôpital's) Rule. That is the usual tool used to evaluate limits when you have the form

0/0 or ∞/∞ or 0*∞

To find the limit of a function as a variable approaches a specific value, you can follow these steps:

Step 1: Substitute the value the variable approaches into the function.
In this case, the variable is u, and it approaches 2. So, substitute 2 for u in the given function.

sqrt(4(2)+1) - 3/(2-2)

Step 2: Simplify the expression.
4(2) equals 8, and 2-2 equals 0. However, dividing by zero is undefined, so we cannot continue simplifying the expression.

Therefore, we cannot directly substitute the value 2 into the function and find the limit. However, we can attempt to find the limit by using algebraic manipulation.

To do this, let's multiply the fraction term by its conjugate (u-2) to eliminate the denominator:

(sqrt(4u+1) - 3) * (u-2) / (u-2)

Expanding the numerator, we get:

(u*sqrt(4u+1) - 2*sqrt(4u+1) - 3u + 6) / (u - 2)

Now, factor out the square root term in the numerator:

(sqrt(4u+1)*(u - 2) - (3u - 6)) / (u - 2)

Simplifying further, we get:

(sqrt(4u+1)*(u - 2) - 3(u - 2)) / (u - 2)

Now, cancel out the common factors of (u - 2) in the numerator and denominator:

(sqrt(4u+1) - 3)

Finally, we substitute the limit variable u with 2 in the simplified expression:

sqrt(4(2)+1) - 3 = sqrt(9) - 3 = 3 - 3 = 0

Therefore, the correct answer to the limit (as u tends to 2) of sqrt(4u+1) - 3/u-2 is 0.