multiple choice 1 point\nthe function f is given by f(x)=sin((x + 1)/x²). which of the following statements…

Answer:

D. I and III only

Explanation:

Step1: Check horizontal asymptote as $x\to\pm\infty$

Find $\lim_{x\to\pm\infty}\sin\left(\frac{x + 1}{x^{2}}\right)$. Let $t=\frac{x + 1}{x^{2}}=\frac{1}{x}+\frac{1}{x^{2}}$. As $x\to\pm\infty$, $\lim_{x\to\pm\infty}t = 0$. Since $\sin$ is continuous, $\lim_{x\to\pm\infty}\sin\left(\frac{x + 1}{x^{2}}\right)=\sin(0)=0$. So $y = 0$ is a horizontal - asymptote.

Step2: Check horizontal asymptote $y = 1$

We know that $\lim_{x\to\pm\infty}\sin\left(\frac{x + 1}{x^{2}}\right)=0\neq1$, so $y = 1$ is not a horizontal asymptote.

Step3: Check vertical asymptote at $x = 0$

As $x\to0$, $\frac{x + 1}{x^{2}}\to+\infty$. The function $\sin\left(\frac{x + 1}{x^{2}}\right)$ oscillates infinitely many times as $x\to0$, and the function is undefined at $x = 0$. So $x = 0$ is a vertical asymptote.