Find limits
Square root x²+ 12-4/x-2 where x approaches to 2
To find the limit, we can substitute the value that x is approaching into the expression:
Let's substitute x = 2 into the expression:
√(2² + 12 - 4)/(2 - 2)
= √(4 + 12 - 4)/0
Now, we can simplify the expression:
= √(12)/0
Since we have a 0 in the denominator, the expression is undefined. Therefore, the limit does not exist.
To find the limit of the expression sqrt(x^2 + 12) - 4 / (x - 2) as x approaches 2, we can begin by substituting 2 into the expression. However, this would result in an undefined expression since we would be dividing by zero.
To address this, we can attempt to simplify the expression before substituting x = 2. We can do this by multiplying the numerator and denominator by the conjugate of sqrt(x^2 + 12) - 4, which is sqrt(x^2 + 12) + 4.
By using the difference of squares formula, we have:
(sqrt(x^2 + 12) - 4) * (sqrt(x^2 + 12) + 4) = (x^2 + 12) - 4^2 = x^2 + 12 - 16 = x^2 - 4
The expression becomes:
((sqrt(x^2 + 12) - 4) * (sqrt(x^2 + 12) + 4)) / ((x - 2) * (sqrt(x^2 + 12) + 4))
Now, by simplifying this expression further:
(x^2 - 4) / (x - 2)
We can now substitute x = 2 into this simplified expression:
(2^2 - 4) / (2 - 2) = 0 / 0
The expression still evaluates to an indeterminate form, which means we need to further simplify it to evaluate the limit. However, since we cannot continue simplifying, we can apply L'Hôpital's rule.
Differentiating the numerator and the denominator with respect to x:
d/dx (x^2 - 4) / d/dx (x - 2)
Simplifying the derivatives:
(2x) / (1) = 2x
Next, we can substitute x = 2 into the differentiated expression:
2 * 2 = 4
Therefore, the limit of the expression as x approaches 2 is equal to 4.