Question #323762

given lim(as x approaches a) for ((x^2+bx+4b)/(x-a))= -6, find the values of a and b.


1
Expert's answer
2022-04-08T13:18:58-0400

In our case, we have an uncertainty of type 0/0.

It follows from the fact that the denominator tends to zero when a tends to zero, and hence the numerator also tends to zero, otherwise the limit would tend to infinity.

x2+bx+4b=0x^2+bx+4b=0 if x=ax=a and due to factor theorem we can write:

x2+bx+4b=(xa)(xc)x^2+bx+4b=(x-a)(x-c) , where cc is some constant. To make limit equal to -6 we should choose c=a+6c=a+6 .

limxax2+bx+4bxa=limxa(xa)(xa6)xa=\displaystyle \lim_{\mathclap{x\to a}} {\frac{x^2+bx+4b}{x-a}}=\lim_{\mathclap{x\to a}} {\frac{(x-a)(x-a-6)}{x-a}}=limxa(xa6)=6\displaystyle\lim_{\mathclap{x\to a}} {(x-a-6)}=-6

x2+bx+4b=(xa)(xa6)x^2+bx+4b=(x-a)(x-a-6)

x2+bx+4b=x2x(2a+6)+a2+6ax^2+bx+4b=x^2-x(2a+6)+a^2+6a

Equating coefficients, we get:

b=(2a+6)b=-(2a+6)

4b=a2+6a4b=a^2+6a

Substituting first equation into the second we will obtain:

4(2a+6)=a2+6a-4(2a+6)=a^2+6a

a2+14a+24=0a^2+14a+24=0

Using Vieta's formulas we can write

a=12a=-12, b=(2a+6)=(2(12)+6)=18b=-(2a+6)=-(2\cdot(-12)+6)=18

a=2a=-2, b=(2(2)+6)=2b=-(2\cdot(-2)+6)=-2

Answer: a=12a=-12 , b=18b=18 or a=2a=-2, b=2b=-2 .


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