I need help with a Calculus I problem, here is the problem:?

Question

Use L'Hopital's rule to evaluate; lim x 1 6x^2-7x^3/2-x^1/2+2 / 6x-6. Please someone explain this so I can understand it..Thank you in advance for your help.

Answer 1

First ensure that the limit as x approaches 1 of the numerator is 0 and the limit as x approaches 1 of the denominator is also 0 so that L'Hopital's rule applies.

Since that is the case, L'Hopital's rule tells us that the limit of this expression is the same as the limit of the expression you get by taking the derivative of the numerator and the derivative of the denominator, if such a limit exists. Doing this gives us the expression (12x - (21/2)x^(1/2) - (1/2)x^-(1/2))/6. Taking the limit as x goes to 1 of this expression gives us a real number, 1/6.