evaluate
lim of x as it approaches infinity for sqrt of x^2 -1/2x+1
x^2 -1/2x+1 = x/2 - 1/4 - 3/(4(2x+1))
as x-->oo, that goes to oo as well.
Discard all of the numerator and denominator except the highest powers. For large x, your function just looks like x/2. Big, big limit.
To evaluate the limit as x approaches infinity for the expression sqrt(x^2 - 1)/2x + 1, we can use some algebraic manipulation.
First, let's simplify the expression inside the square root:
sqrt(x^2 - 1) = sqrt((x^2 - 1)/(x^2)) = sqrt(1 - 1/x^2)
Now, let's divide both the numerator and denominator by x:
sqrt(1 - 1/x^2) = sqrt((1/x^2) - (1/x^4))
As x approaches infinity, both terms (1/x^2 and 1/x^4) approach zero. Therefore, we can simplify the expression further:
sqrt((1/x^2) - (1/x^4)) = sqrt(0 - 0) = sqrt(0) = 0
Now, let's substitute this result back into the original expression:
lim(x→∞) (sqrt(x^2 - 1)/2x + 1) = lim(x→∞) (0/2x + 1) = 0/∞ + 1 = 0 + 1 = 1
Therefore, the limit of sqrt(x^2 - 1)/2x + 1 as x approaches infinity is 1.
To evaluate the limit of the expression sqrt(x^2 - 1) / (2x + 1) as x approaches infinity, we can use some techniques of limit evaluation.
Let's break down the steps:
Step 1: Simplify the expression
We start by simplifying the expression inside the square root. since x approaches infinity, we can ignore the -1 term, as it becomes negligible compared to the x^2 term.
So, sqrt(x^2 - 1) ≈ sqrt(x^2) = x.
The denominator can also be simplified. As x approaches infinity, the 1 term becomes negligible compared to the x term.
So, 2x + 1 ≈ 2x.
Now, the expression becomes x / 2x = 1/2.
Step 2: Write the simplified expression as a limit
Now that we have simplified the expression, we can write it in terms of a limit as x approaches infinity:
lim (x → ∞) (1/2)
Step 3: Evaluate the limit
Since the expression is a constant 1/2, it does not depend on the value of x. Therefore, the limit is equal to the expression itself.
lim (x → ∞) (1/2) = 1/2.
So, the limit of sqrt(x^2 - 1) / (2x + 1) as x approaches infinity is 1/2.