`lim_(x->0) (sqrt(25-x^2)-5)/x` Evaluate the limit, using L’Hôpital’s Rule if necessary.

Given to solve,

`lim_(x->0) (sqrt(25-x^2)-5)/x`

upon Rationalizing numerator we get

=` lim_(x->0) ((sqrt(25-x^2)-5)/x) ((sqrt(25-x^2)+5)/(sqrt(25-x^2)+5))`

=`lim_(x->0) (((sqrt(25-x^2)^2-5^2)/(x(sqrt(25-x^2)+5)))`

=`lim_(x->0) ((((25-x^2)-25)/(x(sqrt(25-x^2)+5)))`

=`lim_(x->0) ((-x^2)/(x(sqrt(25-x^2)+5)))`

=`lim_(x->0) ((-x)/((sqrt(25-x^2)+5)))`

Now plugging the value of `x =0 ` we get

`((-x)/((sqrt(25-x^2)+5)))`

= `((-0)/((sqrt(25-0^2)+5)))`

=`0`

Comments

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