Question #62181

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Feb 4, 2018

Answer:

#lim_(x->0) (1-sqrt(1-x^2))/(3x)=#

Explanation:

We want to find #lim_(x->0) (1-sqrt(1-x^2))/(3x)#

First we plug in the limiting value to see if our limit is indeterminate or not.

#(1-sqrt(1-0^2))/(3(0))=0/0# so the limit is finite.

#lim_(x->0) (1-sqrt(1-x^2))/(3x)=lim_(x->0)((1-sqrt(1-x^2))(1+sqrt(1-x^2)))/((3x)(1+sqrt(1-x^2)))=lim_(x->0)(1-(1-x^2))/((3x)(1+sqrt(1-x^2)))=lim_(x->0)x^2/((3x)(1+sqrt(1-x^2)))=lim_(x->0) x/(3(1+sqrt(1-x^2))#

Now we see that plugging in #0# won't give us an expression which we can't evaluate so we can plug in #0# again

The limit is #0/(3(1+sqrt(1-0^2)))=0#

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