Question

evaluate the following: lim_{x→∞} cos(e^{-x}); options: π/2, 1, dne, 0, -1; question 7 1 pts; evaluate the following: lim_{x→∞} (x² - 1)/x; options: 1, ∞, 0, -∞, -1

Explanation:

Step1: Analyze inner - function limit

As $x\to\infty$, for the function $y = e^{-x}=\frac{1}{e^{x}}$, we have $\lim_{x\to\infty}e^{-x}=\lim_{x\to\infty}\frac{1}{e^{x}} = 0$.

Step2: Evaluate outer - function limit

Let $t = e^{-x}$. Then $\lim_{x\to\infty}\cos(e^{-x})=\lim_{t\to0}\cos(t)$. Since $\cos(t)$ is continuous at $t = 0$, and $\cos(0)=1$, so $\lim_{x\to\infty}\cos(e^{-x}) = 1$.

for Question 7:

Step1: Simplify the function

We have $\lim_{x\to\infty}\frac{x^{2}-1}{x}=\lim_{x\to\infty}(x-\frac{1}{x})$.

Step2: Evaluate the limit of each term

As $x\to\infty$, $\lim_{x\to\infty}x=\infty$ and $\lim_{x\to\infty}\frac{1}{x}=0$. Then $\lim_{x\to\infty}(x - \frac{1}{x})=\lim_{x\to\infty}x-\lim_{x\to\infty}\frac{1}{x}=\infty-0=\infty$.

Answer:

1