Mathematics

Problem

The function

g(x)=\dfrac{(k-3)x^{2}+4x-1}{(k+1)x^{2}-7}

depends on a real parameter

k

. (a) Find the value of

k

for which

\displaystyle \lim_{x \to \infty} g(x)

exists and is equal to

\dfrac{1}{2}

. Show all necessary steps. (b) For the value of

k

obtained in part (a), determine

\displaystyle \lim_{x \to -\infty} g(x)

. State your answer clearly.

Intuitive explanation

When $x \to \infty$ for a rational function, the limit depends on the degrees of the numerator and denominator polynomials. If both have the same degree, the limit is the ratio of their leading coefficients—so you need to match those coefficients to get $\frac{1}{2}$ . The key insight is that the leading coefficient of the numerator is $(k-3)$ and of the denominator is $(k+1)$ , so set up an equation and solve for $k$ . Once you have $k$ , the limit as $x \to -\infty$ follows from the same leading-coefficient ratio, since the behavior at $-\infty$ is identical to that at $+\infty$ for rational functions.

Intuitive explanation

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