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...
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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