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 determine if your substitution and simplification is correct, we need to take a closer look at your expression and the process you followed.
The given expression is g(x) = 2 * ((x - 2) / (sqrt(x) - 2)). You substituted 2 in place of x, resulting in g(2) = 2 * ((2 - 2) / (sqrt(2) - 2)). Simplifying this expression gives you g(2) = 0 / (sqrt(2) - 2), which appears as 0.
However, this approach is incorrect. In general, when finding the limit of a function, you should not substitute the value directly into the function and evaluate it. Instead, you should analyze the behavior of the function as it approaches the given value.
To find the limit of g(x) as x approaches 2 correctly, you can follow these steps:
1. Factor the expression to simplify it:
g(x) = 2 * ((x - 2) / (sqrt(x) - 2))
= 2 * ((x - 2) / (sqrt(x) - 2)) * ((sqrt(x) + 2) / (sqrt(x) + 2))
= 2 * ((x - 2) * (sqrt(x) + 2)) / ((sqrt(x) - 2) * (sqrt(x) + 2))
= 2 * (x - 2)(sqrt(x) + 2) / (x - 4)
2. Cancel out common factors:
g(x) = 2 * (sqrt(x) + 2), for x ≠ 4
3. Now, you can directly substitute the limit value x = 2 into the simplified expression:
lim(x->2) g(x) = lim(x->2) (2 * (sqrt(x) + 2))
= 2 * (sqrt(2) + 2)
Therefore, the correct limit of g(x) as x approaches 2 is 2 * (sqrt(2) + 2).
It's important to note that evaluating expressions by substitution only works when you are finding the value of a function at a specific point, not when finding the limit of a function.