Math / Calculus

Question $\lim _{x \rightarrow 0} x \cot x$

Studdy Solution

STEP 1

1. The limit involves a trigonometric function, specifically the cotangent function.
2. The cotangent function can be expressed as the ratio of cosine and sine functions: $\cot x = \frac{\cos x}{\sin x}$ .
3. The behavior of $\sin x$ and $\cos x$ near $x = 0$ is well-known, and the small-angle approximations for these functions can be utilized.
4. The limit should be evaluated using algebraic manipulation and known limits of trigonometric functions.

STEP 2

1. Express $\cot x$ in terms of sine and cosine.
2. Simplify the expression $x \cot x$ .
3. Use the known limit properties and trigonometric limits to evaluate the limit as $x$ approaches $0$ .

STEP 3

Rewrite $\cot x$ in terms of $\sin x$ and $\cos x$ .
$\cot x = \frac{\cos x}{\sin x}$

STEP 4

Substitute $\cot x$ in the given limit expression.
$\lim _{x \rightarrow 0} x \cot x = \lim _{x \rightarrow 0} x \left( \frac{\cos x}{\sin x} \right)$

STEP 5

Simplify the expression.
$\lim _{x \rightarrow 0} x \left( \frac{\cos x}{\sin x} \right) = \lim _{x \rightarrow 0} \frac{x \cos x}{\sin x}$

STEP 6

Use the limit property $\lim_{x \to 0} \frac{\sin x}{x} = 1$ . Note that $\cos x$ approaches $1$ as $x$ approaches $0$ .
$\lim _{x \rightarrow 0} \frac{x \cos x}{\sin x} = \lim _{x \rightarrow 0} \left( \frac{x}{\sin x} \cdot \cos x \right)$

STEP 7

Separate the limit into two parts and evaluate each part.
$\lim _{x \rightarrow 0} \left( \frac{x}{\sin x} \cdot \cos x \right) = \left( \lim _{x \rightarrow 0} \frac{x}{\sin x} \right) \cdot \left( \lim _{x \rightarrow 0} \cos x \right)$

STEP 8

Evaluate each limit individually.
$\left( \lim _{x \rightarrow 0} \frac{x}{\sin x} \right) = 1$ $\left( \lim _{x \rightarrow 0} \cos x \right) = 1$

STEP 9

Combine the results.
$\left( \lim _{x \rightarrow 0} \frac{x}{\sin x} \right) \cdot \left( \lim _{x \rightarrow 0} \cos x \right) = 1 \cdot 1 = 1$
Solution: $\lim _{x \rightarrow 0} x \cot x = 1$

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