Solve lim x-> 25 (5 - sqrt x)/(x - 25) using two methods.

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We have to determine `lim_(x-> 25)(5 - sqrt x)/(x - 25)`

If we substitute x = 25 we get the result 0/0 which is indeterminate.

One way of finding the limit is to multiply the numerator and denominator by `5 + sqrt x`

=> `lim_(x->25)((5-sqrt x)(5+sqrtx))/((x-25)(5+sqrt x))`

`(5 - sqrt x)(5 + sqrt x) = 25 - x`

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

=> `lim_(x-> 25)-1/(5+sqrt x)`

substitute x = 25

=> -1/(5 + 5) = -1/10

Another way to find the limit is to use the fact a^2 - b^2 = (a - b)(a + b)

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

`lim_(x->25) (5 - sqrt x)/(x - 25)`

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

=> `lim_(x-> 25)-1/(sqrt x + 5)`

=> -1/10

The required limit is -1/10

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