limit(2*(square root x) - (square root x+2)) x approaching infinity
To evaluate the limit of the expression 2√x - √(x+2) as x approaches infinity, we can use a limit property that states that the limit of a sum (or difference) is equal to the sum (or difference) of the limits.
First, let's evaluate the limits of the individual terms separately:
1. The limit of 2√x as x approaches infinity:
As x approaches infinity, the square root of x also approaches infinity. Multiplying it by 2 will not change this result, so 2√x approaches infinity as well.
2. The limit of √(x+2) as x approaches infinity:
As x approaches infinity, the value inside the square root, x+2, also approaches infinity. Taking the square root of a larger number will result in a larger number, so √(x+2) also approaches infinity.
Now, we can combine the limits of the individual terms:
lim(x→∞) [2√x - √(x+2)] = [lim(x→∞) 2√x] - [lim(x→∞) √(x+2)]
Since both limits approach infinity, we have:
infinity - infinity
This result is an indeterminate form, meaning we cannot determine the exact value of the limit from this expression alone. To evaluate it further, we can use algebraic techniques such as rationalizing the expression or manipulating it in some way.
However, based on the given expression, we can conclude that the limit of 2√x - √(x+2) as x approaches infinity is undefined or indeterminate.