evaluate the limit as x approaches infinity for the
absolute value of (the square root of xsquared+x) -x
i think the answer is 1/2 from my calculator but I have to show how i got it algebricallly and have no clue what to do
To evaluate the limit as x approaches infinity for the absolute value of (sqrt(x^2+x) - x), we'll need to simplify the expression and apply some algebraic techniques.
Step 1: Simplify the expression inside the absolute value |...|.
Let's start by simplifying the expression by rationalizing the numerator:
(sqrt(x^2+x) - x) * (sqrt(x^2+x) + x) / (sqrt(x^2+x) + x)
= (x^2 + x - x^2) / (sqrt(x^2+x) + x)
= x / (sqrt(x^2+x) + x)
Step 2: Apply division with the highest power technique.
To apply this technique, we can divide both the numerator and denominator by the highest power of x, which is x:
x / (sqrt(x^2+x) + x)
= x/x * (1 / (sqrt(x^2+x)/x + x/x))
= 1 / (sqrt(1 + 1/x) + 1)
Step 3: Evaluate the limit as x approaches infinity.
Now, as x approaches infinity, the term 1/x goes to zero, and we are left with:
lim(x→∞) 1 / (sqrt(1 + 1/x) + 1)
= 1 / (sqrt(1 + 0) + 1)
= 1 / (sqrt(1) + 1)
= 1 / (1 + 1)
= 1/2
So, the limit as x approaches infinity for the given expression is 1/2.