The problem is lim --> 2 for g(x) which is 2((x-2/(squareroot x)-2)) I just substituted 2 in for x and 2((2-2)/(squareroot 2)-2) = 0 This doesn't look right. It seems like I'd need to try something different.
Can someone tell me if it's right or wrong? And if there's a different approach?
To evaluate the limit, we need to simplify the expression and then substitute the value of x.
Given g(x) = 2((x-2)/(sqrt(x)-2)), you substituted 2 in for x, which gave you 2((2-2)/(sqrt(2)-2)) = 0.
However, upon closer inspection, we can see that the expression is not properly simplified. Let's break it down:
g(x) = 2((x-2)/(sqrt(x)-2))
To simplify, we can multiply the numerator and denominator by the conjugate of sqrt(x)-2, which is sqrt(x) + 2. This will eliminate the square root from the denominator:
g(x) = 2((x-2)/(sqrt(x)-2)) * ((sqrt(x)+2)/(sqrt(x)+2))
Multiplying the numerators and the denominators, we get:
g(x) = 2(x-2)(sqrt(x) + 2) / ((sqrt(x)-2)(sqrt(x)+2))
Now, we can simplify further:
g(x) = 2(x-2)(sqrt(x) + 2) / (x - 4)
Now, let's substitute x = 2 into this expression:
g(2) = 2(2-2)(sqrt(2) + 2) / (2 - 4)
= 0 / (-2)
= 0
So, after simplifying the expression correctly and substituting x = 2, we find that g(x) evaluates to 0, which matches your previous result. Therefore, your initial calculation was indeed correct.
There is no need for a different approach as long as we simplify the expression properly before substituting the value of x.