Question

If k and b are constant such that lim x approach infinity (kx+b-(x^3+1)/(x^2+1)=0. Find the values of k and b

Answer 1

Combining the terms into one fraction, we have

`kx + b - (x^3+1)/(x^2+1) = ((k-1)x^3 + bx^2 + kx + b - 1)/(x^2+1)`

If this converges to 0 as
`x\to\infty`, then the degree of the numerator must be smaller than the degree of the denominator.

To ensure this, take
`k=1` and
`b=0`. This eliminates the cubic and quadratic terms in the numerator, and we do have

`\displaystyle \lim_(x\to\infty) (x - 1)/(x^2 + 1) = \lim_(x\to\infty) \frac{\frac1x - \frac1{x^2}}{1 + \frac1{x^2}} = 0`

Alternatively, we can compute the quotient and remainder of the rational expression.

`(x^3+1)/(x^2+1) = x - (x-1)/(x^2+1)`

Then in the limit, we have

`\displaystyle \lim_(x\to\infty) \left(kx + b - x + (x-1)/(x^2+1)\right) = (k-1) \lim_(x\to\infty) x + b = 0`

Both terms on the left vanish if
`k=1` and
`b=0`.