find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise…

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)\n$$\\lim_{x \\to \\infty} 3 \\cos(x)$$\n

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)\n$$\\lim_{x \\to \\infty} 3 \\cos(x)$$\n

Answer

Explanation:

Step1: Analyze the behavior of $\cos(x)$ as $x\to\infty$

The cosine function $\cos(x)$ is a periodic function with period $2\pi$. Its values oscillate between $- 1$ and $1$ for all real values of $x$. That is, $-1\leqslant\cos(x)\leqslant1$ for all $x\in R$.

Step2: Analyze the behavior of $3\cos(x)$ as $x\to\infty$

Multiply the inequality $-1\leqslant\cos(x)\leqslant1$ by $3$. We get $-3\leqslant3\cos(x)\leqslant3$. As $x\to\infty$, the function $y = 3\cos(x)$ does not approach a single, finite value. For example, when $x = 2n\pi$ ($n\in N$), $3\cos(2n\pi)=3$; when $x=(2n + 1)\pi$ ($n\in N$), $3\cos((2n + 1)\pi)=-3$.

Answer:

DNE