I need to solve:
Lim as x is approaching 1/4, for (4x-1)/(1/ ([square root of x] minus 2)).
Help would be greatly appreciated!!!
It's trivial using L'Hopital's rule.
Answer:
-64
Correction, the answer is -1.
what is l'hospital rule?
L'Hôpital's Rule, named after French mathematician Guillaume de l'Hôpital, is a method for evaluating limits of indeterminate forms that involve fractions of functions. The rule states that if the limit of the ratio of two functions f(x) and g(x) as x approaches a certain value is an indeterminate form (such as 0/0 or ∞/∞), then the limit of the ratio can often be found by differentiating both the numerator and the denominator of the original fraction and then taking the limit of the resulting fraction.
To use L'Hôpital's Rule, follow these steps:
1. Determine if the limit you're trying to evaluate gives an indeterminate form (such as 0/0 or ∞/∞).
2. Rewrite the limit as a fraction of two functions, with the numerator and denominator both being functions of x.
3. Differentiate both the numerator and the denominator separately with respect to x.
4. Take the limit of the resulting fraction as x approaches the given value.
5. If the limit is still indeterminate, repeat the process of differentiating the numerator and denominator until you find a limit that is not indeterminate.
Now, let's solve the given problem using L'Hôpital's Rule.
We have the limit as x approaches 1/4 of (4x-1)/(1/([√x] - 2)).
First, let's simplify the expression in the denominator. The square root of x can be written as x^(1/2).
So, we can rewrite the expression as (4x-1)/(1/(x^(1/2) - 2)).
Now, we can apply L'Hôpital's Rule. We differentiate the numerator and denominator:
The derivative of 4x-1 is 4,
The derivative of 1/(x^(1/2) - 2) is -1/((x^(1/2) - 2)^2) * (1/2)x^(-1/2).
Taking the limit as x approaches 1/4 again, we have:
lim (x->1/4) [4 / (-1/((x^(1/2) - 2)^2) * (1/2)x^(-1/2))]
Now, let's simplify this expression further:
lim (x->1/4) [4 * (-2)^2 * (1/2) * x^(-1/2)]
Simplifying the exponent and the constants:
lim (x->1/4) [4 * 2 * (1/2) * x^(-1/2)]
= lim (x->1/4) [ 4 * x^(-1/2) ]
= 4 * (1/(√(1/4)))
= 4/(1/2)
= 4 * 2
= 8
Therefore, the limit as x approaches 1/4 of (4x-1)/(1/([√x] - 2)) is 8.