What is lim_(x-> 0)(sqrt(x+5) - sqrt 5)/x

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The limit `lim_(x-> 0)(sqrt(x+5) - sqrt 5)/x` has to be determined.

There are two ways of doing this.

For x = 0, `(sqrt(x+5) - sqrt 5) = (sqrt 5 - sqrt 5) = 0` , this makes the expression `(sqrt(x+5) - sqrt 5)/x = 0/0` which is indeterminate. l'Hospital's rule can be used now and the numerator and denominator substituted with their derivatives.

Doing this gives `lim_(x->0) 1/(2*sqrt(x+5))`

At x = 0, the expression is equal to `1/(2*sqrt(x+5)) = 1/(2*sqrt 5)`

Another way of determining the limit is by using factorization and the formula x^2 - y^2 = (x - y)(x + y)

`lim_(x-> 0)(sqrt(x+5) - sqrt 5)/x`

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

= `lim_(x-> 0)(sqrt(x+5) - sqrt 5)/((sqrt(x + 5))^2 - (sqrt5)^2)`

= `lim_(x-> 0)(sqrt(x+5) - sqrt 5)/((sqrt(x + 5) + sqrt5)*(sqrt(x + 5) - sqrt5))`

= `lim_(x-> 0)1/((sqrt(x + 5) + sqrt5))`

At x = 0 this is equal to `1/(2*sqrt5)`

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