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 given limit, let's first rewrite the expression:
g(x) = 2((x-2)/(√x-2))
Now, to determine whether substituting 2 for x in this expression is correct or not, we need to check if the limiting behavior of g(x) as x approaches 2 can be determined directly from the expression itself.
The denominator of g(x) contains a square root expression (√x), which becomes undefined for x ≤ 0. Since we are approaching x = 2, which is a positive number, we should be fine. However, substituting the value 2 for x in the expression results in division by zero, which is undefined.
Therefore, substituting 2 for x doesn't yield the correct answer because it leads to an indeterminate form. To evaluate the limit correctly, you may need to try a different approach.
One possible approach is to simplify the expression to get rid of the indeterminate form:
g(x) = 2((x-2)/(√x-2)) × (√x+2)/(√x+2)
Now, we can combine the fractions to obtain:
g(x) = 2(x-2)/(√x+2)
Next, we can try evaluating the limit using this simplified expression. We can either directly substitute 2 for x or apply more advanced techniques such as L'Hôpital's Rule or factorization to determine the limit.
By substituting x = 2 into the simplified expression, we get:
g(2) = 2(2-2)/(√2+2) = 0/(2+2) = 0/4 = 0
Therefore, the limit of g(x) as x approaches 2 is indeed 0.