f x = e α x − e x − x x 2 , x ≠ 0 3 2 , x = 0 Find the value of α for which the function f is continuous.

Step-by-step solution

For this function to be continuous lim x → 0 e α x − e x − x x 2 = 3 2 , it is  0 0   form Now, applying L'Hospital rule, lim x → 0 α e α x − e x − 1 2 x = α e α 0 − e 0 − 1 2 x Now, as denominator gives the value 0 . In order for the limit to exist, the value of numerator, α e α 0 − e 0 − 1 = 0 ⇒ α = 2