Evaluate lim x-->0 sqrt(3+x) - sqrt(3) / x

Expert Answers

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lim f(x) = lim [sqrt(3+x) - sqrt(3)]/x    when x--> 0

By substitution:

lim f(x) = 0/0

Now let us multiply and divide by (sqrt(3+x) + sqrt(3)

==> lim [(sqrt(3+x)- sqrt(3))*(sqrt(3+x) + sqrt(3)]/x(sqrt(3+x)+sqrt3).

= lim (3+x - 3)/ x(sqrt(3+x) + sqrt(3)  

= lim x/x(sqrt(x+3)+sqrt3)

= lim 1/(sqrt(3+x) + sqrt3)

Then,

lim f(x) when x--> 0  = 1/(sqrt3+0) + sqrt3)

                                  = 1/2sqrt3 = sqrt3/6

Then lim f(x) = sqrt3/6

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