1 Answer

Brewmanz · Answer 1 · 2023-12-11T20:42:39+0000

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 .

Please log in or register to add a comment.