Question

If `f(x)=-(25-x^2)^(1/2)`, find `lim_(x->1)(f(x)-f(1))/(x-1)`?

Answer 1

`lim_(x->1)(f(x)-f(1))/(x-1)=1/(2sqrt6)`

Explanation:

As `f(x)=-(25-x^2)^(1/2)=-sqrt(25-x^2)`, and we have

`f(1)=-(25-1)^(1/2)=-sqrt24`

then `lim_(x->1)(f(x)-f(1))/(x-1)`

= `lim_(x->1)-(sqrt(25-x^2)-sqrt24)/(x-1)`

= `lim_(x->1)-(sqrt(25-x^2)-sqrt24)/(x-1)xx(sqrt(25-x^2)+sqrt24)/(sqrt(25-x^2)+sqrt24)`

= `lim_(x->1)-(25-x^2-24)/((x-1)(sqrt(25-x^2)+sqrt24))`

= `lim_(x->1)-(1-x^2)/((x-1)(sqrt(25-x^2)+sqrt24))`

= `lim_(x->1)(1+x)/(sqrt(25-x^2)+sqrt24)`

= `lim_(x->1)(1+1)/(sqrt(25-1)+sqrt24)`

= `2/(2sqrt24)`

= `1/sqrt24=1/(2sqrt6)`