Question

evaluate the limit when X is 0 1 - X raise to power n minus one by X ​

Answer 1

Answer:

The limit is  -n

Step-by-step explanation:

Using the binomial theorem:

(1-x)ⁿ = 1 - nx + [ terms with x² and higher degrees ]

=> (1-x)ⁿ - 1 = -nx + [ terms with x² and higher degrees ]

=> ( (1-x)ⁿ - 1 ) / x = -n + [ terms with x¹ and higher degrees ]

As x tends to 0, the terms with x¹, x², x³ and higher degrees, all tend to 0.  So all that remains is the constant -n.

Thus

lim (as x -> 0)  ( (1-x)ⁿ - 1 ) / x   =  -n