The function f is given by the formula: f(x)=(3x^3+5x^2+7x+18)/(x+2) when x is l

Question

The function f is given by the formula: f(x)=(3x^3+5x^2+7x+18)/(x+2) when x is less than or equal to 2. and by the formula f(x)=1x^2+1x+a when -2 is less than or equal to x. What value must be chosen for a in order to make this function continuous at -2?

Explanation / Answer

find the function value as x--> -2 ............. so we have................for x is less than or equal to 2 lim x--> -2 f(x) = lim x--> -2 (3x^3+5x^2+7x+18)/(x+2) = 0 / 0 which is indeterminate form .......................so using L'Hospital rule .... lim x--> -2 (9x^2 + 10x + 7 +0) /(1 + 0) = 23 ----------->(1) ................ now for -2 is less than or equal to x,....we have...... lim x--> -2 (1x^2+1x+a) = 2 + a --------->(2)............ we have to make this function continuous at -2............so... from (1) and (2)........23 = 2+a ==> a = 21

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