Question

If f(x) = 2ax + b/x-1, lim x→0 f(x) = -3, lim x→∞ f(x)=4, prove that: f(2)= 11. ​

Answer 1

If

`f(x) = (2ax+b)/(x-1)`

Note that

`f(2) = (4a+b)/(2-1) = 4a+b`

From the given information we have

`\displaystyle \lim_(x\to0) f(x) = \lim_(x\to0) (2ax+b)/(x-1) = (0+b)/(0-1) = -b = -3 \\\\ ~~~~ \implies b=3`

and

`\displaystyle \lim_(x\to\infty) f(x) = \lim_(x\to\infty) (2ax+b)/(x-1) = \lim_(x\to\infty) (2a+\frac bx)/(1-\frac1x) = (2a+0)/(1-0) = 2a = 4 \\\\ ~~~~ \implies a=2`

It follows that

`f(2) = 4\cdot2+3 = 11`

as required.