Evaluate the limit

Question

Lim_x→1 (√x²+6x)-(√7) / (x-1)

Answer 1

The limit has the indeterminate form 0/0

 

Multiply the numerator and denominator by √(x²+6x) + √7 to obtain:

 

    lim_x→1 [(x²+6x-7) / ((x-1)(√(x²+6x) + √7))]

 

= lim_x→1 [(x+7)(x-1) / ((x-1)(√(x²+6x) + √7))]

 

= lim_x→1 [(x+7) / (√(x²+6x) + √7)] = 8 / (2√7) = 4 / √7

 

                                                                        = 4√7 / 7