Question

a) given f(x)= { 2x²+1, x<-2
               { ax-1, x≥-2
Determine the value of a if 
lim f(x) exists.
x -2

b) find the value of a such
 that lim 2x²-ax-14/x²-2x-3 
      x -4

...esists. hence determine the value of the limit

Answer 1

a)

$$\lim\limits_{x\to-2^{-}}f(x)=2(-2)^2+1=9$$

$$\lim\limits_{x\to-2^{+}}f(x)=a(-2)-1=-2a-1$$

$$\lim\limits_{x\to-2^{-}}f(x)=\lim\limits_{x\to-2^{+}}f(x)=\lim\limits_{x\to-2}f(x)$$

$$9=-2a-1$$

$$a=5$$

b)

$$\lim\limits_{x\to-1}\dfrac{2x^2-ax-14}{x^2-2x-3}=\lim\limits_{x\to-2}\dfrac{2x^2-ax-14}{(x+1)(x-3)}$$

$$2(-1)^2-a(-1)-14=0=>a=12$$

$$2x^2-ax-14=2x^2-12x-14$$

$$=2(x^2-6x-7)=2(x+1)(x-7)$$

$$\lim\limits_{x\to-1}\dfrac{2x^2-12x-14}{x^2-2x-3}=\lim\limits_{x\to-2}\dfrac{2(x+1)(x-7)}{(x+1)(x-3)}$$

$$=\lim\limits_{x\to-2}\dfrac{2(x-7)}{x-3}=\dfrac{2(-1-7)}{-1-3}=4$$