lim x-->0+ (sqrt((1/x)+2)-sqrt(1/x))
I used my calculator and got 0!
using l'Hospital's Rule, the limit is the same as that of
2√x / √(1+2x) --> 0
To find the limit of the given expression, we can start by simplifying it.
Let's rewrite the expression as follows:
lim x→0+ (sqrt((1/x)+2) - sqrt(1/x))
Since the limit is approaching 0+ (from the right side), we can break down the limit expression into two separate limits and then evaluate them individually:
lim x→0+ sqrt((1/x)+2) - lim x→0+ sqrt(1/x)
Let's simplify each limit separately:
1. For the first limit:
lim x→0+ sqrt((1/x)+2)
To evaluate this limit, we need to rationalize the expression. Multiply both the numerator and denominator by the conjugate of the denominator, which is sqrt((1/x)+2) + sqrt(1/x).
lim x→0+ ((sqrt((1/x)+2))*(sqrt((1/x)+2) + sqrt(1/x)))
lim x→0+ ((1/x)+2 + sqrt(1/x)*sqrt((1/x)+2))
Next, simplify the expression inside the square root:
lim x→0+ ((1/x)+2 + sqrt((1/x)*((1/x)+2)))
lim x→0+ ((1/x)+2 + sqrt(1/x^2 + 2/x))
Since x is approaching zero from the right side, (1/x) and (2/x) will approach positive infinity, so we can ignore them. Also, since (1/x^2) is a very small value, we can consider it as negligible compared to 2/x.
Thus, the expression simplifies to:
lim x→0+ (2 + sqrt(2/x))
As x approaches zero from the right side, 2/x goes to infinity. In this case, the expression inside the square root approaches positive infinity, so the square root evaluates to infinity as well. Therefore, the first limit is positive infinity.
2. For the second limit:
lim x→0+ sqrt(1/x)
As x approaches zero from the right side, 1/x goes to infinity. As a result, the expression inside the square root also approaches infinity. Therefore, the second limit is also positive infinity.
Now, let's evaluate the original limit expression by subtracting the two separate limits:
lim x→0+ (sqrt((1/x)+2) - sqrt(1/x))
= (positive infinity) - (positive infinity)
When both limits are positive infinity, we cannot determine the exact value of the difference. Thus, the answer to the limit expression is indeterminate.
In summary, the limit of the given expression, lim x→0+ (sqrt((1/x)+2) - sqrt(1/x)), is indeterminate.