Answer all 3 and show all work 1) Lim square root * cos 1/ x^2 x > 0 + 2) Find I

Question

Answer all 3  and show all work

1) Lim    square root  * cos 1/ x^2

    x > 0 +

2) Find Intervals on which f(x) is continuous.

     f(x) = x+1/2 e^x -8

3) Find K such that f(x) is continuous everywhere

   f(x) =  x^2-2x/x^2-4        x cannot equal 2

           

              K                         x=2

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Answer all 3 and show all work Lim square root * cos 1/ x^2 x > 0 + Find Intervals on which f(x) is continuous. f(x) = x+1/2 e^x -8 Find K such that f(x) is continuous everywhere f(x) = x^2-2x/x^2-4 x cannot equal 2

Explanation / Answer

(1)

Lim x-> 0 sqrt(x) cos 1/ x^2

= 0 * Lim x-> 0 cos ( 1/ x^2 )

= 0

(2)

f(x) is continuous when 2e^x - 8 not = 0

=> 2e^x not = 8

=>e^x not = 4

=> x not = ln4

so it is continuous on all values except at x = ln 4

so interval notation is

( - inf , ln 4 ) U ( ln 4 , inf )

(3)

Lim x-> 2 - f (x) = Lim x->2 + f(x)

=> Lim x->2   (x^2 - 2x ) / (x^2 - 4 ) = Lim x->2 k

=> Lim x->2 x(x-2) / (x+2)(x-2) = k

=> Lim x->2 x / ( x +2 ) = k

=> 2 / 4 = k

=> k = 1/2 or 0.5

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