let f be the function defined by f(x) = (2x + 3)/(x + 1). which of the following statements are true? i. the…

let f be the function defined by f(x) = (2x + 3)/(x + 1). which of the following statements are true? i. the graph of f has a horizontal asymptote at y = 2 because lim f(x) = 2 as x→∞. ii. the graph of f has a horizontal asymptote at y = 2 because lim f(x) = 2 as x→ -∞. iii. the graph of f has a vertical asymptote at x = -1 because lim f(x) = ∞ as x→ -1⁺. a i only b iii only c i and ii only d i, ii, and iii

let f be the function defined by f(x) = (2x + 3)/(x + 1). which of the following statements are true? i. the graph of f has a horizontal asymptote at y = 2 because lim f(x) = 2 as x→∞. ii. the graph of f has a horizontal asymptote at y = 2 because lim f(x) = 2 as x→ -∞. iii. the graph of f has a vertical asymptote at x = -1 because lim f(x) = ∞ as x→ -1⁺. a i only b iii only c i and ii only d i, ii, and iii

Answer

Explanation:

Step1: Find $\lim_{x\rightarrow\infty}f(x)$

Divide numerator and denominator by $x$: $\lim_{x\rightarrow\infty}\frac{2x + 3}{x+1}=\lim_{x\rightarrow\infty}\frac{2+\frac{3}{x}}{1+\frac{1}{x}}$. As $x\rightarrow\infty$, $\frac{3}{x}\rightarrow0$ and $\frac{1}{x}\rightarrow0$. So $\lim_{x\rightarrow\infty}\frac{2+\frac{3}{x}}{1+\frac{1}{x}} = 2$.

Step2: Find $\lim_{x\rightarrow-\infty}f(x)$

Again, divide numerator and denominator by $x$: $\lim_{x\rightarrow-\infty}\frac{2x + 3}{x + 1}=\lim_{x\rightarrow-\infty}\frac{2+\frac{3}{x}}{1+\frac{1}{x}}$. As $x\rightarrow-\infty$, $\frac{3}{x}\rightarrow0$ and $\frac{1}{x}\rightarrow0$. So $\lim_{x\rightarrow-\infty}\frac{2+\frac{3}{x}}{1+\frac{1}{x}}=2$. This means $y = 2$ is a horizontal - asymptote.

Step3: Find $\lim_{x\rightarrow - 1^{+}}f(x)$

As $x\rightarrow - 1^{+}$, the numerator $2x + 3\rightarrow2\times(-1)+3=1$ and the denominator $x + 1\rightarrow0^{+}$. So $\lim_{x\rightarrow - 1^{+}}\frac{2x + 3}{x + 1}=\infty$, and $x=-1$ is a vertical asymptote.

Answer:

D. I, II, and III