Evaluate lim x→∞
2x/(√(x^2+3)-4)
in words 2x divide by(root(xsq +3) -4)
limit is the same for
2/(x/√(x^2+3)) = 2√(x^2+3)/x = 2x/x = 2
To evaluate the limit as x approaches infinity of the given expression:
lim x→∞ (2x/(√(x^2+3)-4)),
we need to first simplify the expression by multiplying both the numerator and the denominator by the conjugate of the denominator expression, which is √(x^2+3) + 4. This will allow us to simplify the expression and apply limit properties.
So, let's multiply the numerator and denominator by the conjugate:
lim x→∞ (2x/(√(x^2+3)-4)) * ((√(x^2+3)+4)/(√(x^2+3)+4)).
Now, we can simplify the expression by applying the distributive property and canceling out the difference of squares terms:
lim x→∞ ((2x(√(x^2+3)+4))/((x^2+3) - 16)).
Simplifying further, we can remove the square root from the numerator by multiplying the top and bottom by its conjugate:
lim x→∞ (((2x(√(x^2+3)+4))/((x^2+3) - 16)) * ((√(x^2+3)+4)/ (√(x^2+3)+4))).
This simplifies to:
lim x→∞ ((2x(x^2+3 + 8√(x^2+3) + 16))/ (x^2 - 13)).
Now, we can divide every term in the numerator by x^2 (the highest power of x):
lim x→∞ (2 + (8√(x^2+3))/x^2 + 16/x^2)/(1 - 13/x^2).
As x approaches infinity, the terms (8√(x^2+3))/x^2 and 16/x^2 both go to zero, since the highest power of x in the denominators dominates the terms. Therefore, we are left with:
lim x→∞ (2 + 0 + 0)/(1 - 0).
Simplifying further:
lim x→∞ (2/1) = 2.
Hence, the limit of the expression as x approaches infinity is 2.