Math / Calculus

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

STEP 1

1. We are given the function $\frac{4x}{x + \cos(x)}$ .
2. We need to find the limit as $x$ approaches negative infinity.

STEP 2

1. Analyze the behavior of the numerator and the denominator as $x \rightarrow -\infty$ .
2. Simplify the expression to identify dominant terms.
3. Evaluate the limit of the simplified expression as $x \rightarrow -\infty$ .

STEP 3

Consider the behavior of the numerator $4x$ as $x \rightarrow -\infty$ . The numerator $4x$ tends towards negative infinity.

STEP 4

Consider the behavior of the denominator $x + \cos(x)$ as $x \rightarrow -\infty$ . The term $x$ tends towards negative infinity, while $\cos(x)$ oscillates between $-1$ and $1$ .

STEP 5

To simplify the expression, factor out $x$ from the denominator:
$\frac{4x}{x + \cos(x)} = \frac{4x}{x(1 + \frac{\cos(x)}{x})}$
This simplifies to:
$\frac{4x}{x(1 + \frac{\cos(x)}{x})} = \frac{4}{1 + \frac{\cos(x)}{x}}$

STEP 6

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:
$\boxed{4}$

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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

STEP 1

1. We are given the function $\frac{4x}{x + \cos(x)}$ .
2. We need to find the limit as $x$ approaches negative infinity.

STEP 2

1. Analyze the behavior of the numerator and the denominator as $x \rightarrow -\infty$ .
2. Simplify the expression to identify dominant terms.
3. Evaluate the limit of the simplified expression as $x \rightarrow -\infty$ .

STEP 3

Consider the behavior of the numerator $4x$ as $x \rightarrow -\infty$ . The numerator $4x$ tends towards negative infinity.

STEP 4

Consider the behavior of the denominator $x + \cos(x)$ as $x \rightarrow -\infty$ . The term $x$ tends towards negative infinity, while $\cos(x)$ oscillates between $-1$ and $1$ .

STEP 5

To simplify the expression, factor out $x$ from the denominator:
$\frac{4x}{x + \cos(x)} = \frac{4x}{x(1 + \frac{\cos(x)}{x})}$
This simplifies to:
$\frac{4x}{x(1 + \frac{\cos(x)}{x})} = \frac{4}{1 + \frac{\cos(x)}{x}}$

STEP 6

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

Studdy Solution

STEP 1

1. We are given the function 4xx+cos⁡(x) \frac{4x}{x + \cos(x)} x+cos(x)4x​.2. We need to find the limit as x x x approaches negative infinity.

STEP 2

1. Analyze the behavior of the numerator and the denominator as x→−∞ x \rightarrow -\infty x→−∞.2. Simplify the expression to identify dominant terms.3. Evaluate the limit of the simplified expression as x→−∞ x \rightarrow -\infty x→−∞.

STEP 3

Consider the behavior of the numerator 4x 4x 4x as x→−∞ x \rightarrow -\infty x→−∞. The numerator 4x 4x 4x tends towards negative infinity.

STEP 4

Consider the behavior of the denominator x+cos⁡(x) x + \cos(x) x+cos(x) as x→−∞ x \rightarrow -\infty x→−∞. The term x x x tends towards negative infinity, while cos⁡(x) \cos(x) cos(x) oscillates between −1-1−1 and 111.

STEP 5

STEP 6

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

1. We are given the function $\frac{4x}{x + \cos(x)}$ .
2. We need to find the limit as $x$ approaches negative infinity.

1. Analyze the behavior of the numerator and the denominator as $x \rightarrow -\infty$ .
2. Simplify the expression to identify dominant terms.
3. Evaluate the limit of the simplified expression as $x \rightarrow -\infty$ .

Consider the behavior of the numerator $4x$ as $x \rightarrow -\infty$ . The numerator $4x$ tends towards negative infinity.

Consider the behavior of the denominator $x + \cos(x)$ as $x \rightarrow -\infty$ . The term $x$ tends towards negative infinity, while $\cos(x)$ oscillates between $-1$ and $1$ .