Question

Find limit as x approaches 3 of the quotient of the quantity the square root of the quantity 2 times x plus 3 minus the square root of 3 times x and the quantity x squared minus 3 times x . You must show your work or explain your work in words. (10 points)

Answer 1

-1/18

Explanation:

The function evaluates to 0/0 at x=3, so L'Hopital's rule applies. The ratio of numerator and denominator derivatives is ...

numerator derivative: 1/√(2x+3) -3/(2√(3x)) . . . . = 1/3 -1/2 = -1/6 at x=3

denominator derivative: 2x-3 . . . = 3 at x=3

(numerator derivative)/(denominator derivative) = (-1/6)/3 = -1/18

The limit of the expression as x→3 is -1/18.