QuestionFind the limit, if it exists. (If an answer does not exist, enter DNE.) $$\lim _{x \rightarrow-\infty} \frac{4 x}{x+\cos (x)}$$

Studdy Solution
Evaluate the limit of the simplified expression as $x \rightarrow -\infty$:
Since $\frac{\cos(x)}{x}$ approaches $0$ as $x \rightarrow -\infty$ (because $\cos(x)$ is bounded and $x$ tends to negative infinity), the expression becomes:
$$\lim_{x \rightarrow -\infty} \frac{4}{1 + \frac{\cos(x)}{x}} = \frac{4}{1 + 0} = 4$$
The limit is:
$$\boxed{4}$$