Evaluate limit as x approaches 0 of 2/((sin(x))/x)+(1-cos(x))/x+(tan(x))/x

$lim_{x \to 0} \frac{2}{\frac{\sin (x)}{x}} + \frac{1 - \cos (x)}{x} + \frac{\tan (x)}{x}$

Step 1

Evaluate the limit.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} \frac{2}{\frac{\sin (x)}{x}} + lim_{x \to 0} \frac{1 - \cos (x)}{x} + lim_{x \to 0} \frac{\tan (x)}{x}$

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 0} \frac{1}{\frac{\sin (x)}{x}} + lim_{x \to 0} \frac{1 - \cos (x)}{x} + lim_{x \to 0} \frac{\tan (x)}{x}$

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$2 \frac{lim_{x \to 0} 1}{lim_{x \to 0} \frac{\sin (x)}{x}} + lim_{x \to 0} \frac{1 - \cos (x)}{x} + lim_{x \to 0} \frac{\tan (x)}{x}$

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$2 \frac{1}{lim_{x \to 0} \frac{\sin (x)}{x}} + lim_{x \to 0} \frac{1 - \cos (x)}{x} + lim_{x \to 0} \frac{\tan (x)}{x}$

Step 2

Since $\cos (x) \leq \frac{\sin (x)}{x} \leq 1$ and $lim_{x \to 0} \cos (x) = lim_{x \to 0} 1 = 1$ , apply the squeeze theorem.

$2 (\frac{1}{1}) + lim_{x \to 0} \frac{1 - \cos (x)}{x} + lim_{x \to 0} \frac{\tan (x)}{x}$

Step 3

Since $0 \leq \frac{1 - \cos (x)}{x} \leq \frac{x}{2}$ and $lim_{x \to 0} 0 = lim_{x \to 0} \frac{x}{2} = 0$ , apply the squeeze theorem.

$2 (\frac{1}{1}) + 0 + lim_{x \to 0} \frac{\tan (x)}{x}$

Step 4

Apply L\'Hospital\'s rule.

Evaluate the limit of the numerator and the limit of the denominator.

Take the limit of the numerator and the limit of the denominator.

$2 (\frac{1}{1}) + 0 + \frac{lim_{x \to 0} \tan (x)}{lim_{x \to 0} x}$

Evaluate the limit of the numerator.

Move the limit inside the trig function because tangent is continuous.

$2 (\frac{1}{1}) + 0 + \frac{\tan (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$2 (\frac{1}{1}) + 0 + \frac{\tan (0)}{lim_{x \to 0} x}$

The exact value of $\tan (0)$ is $0$ .

$2 (\frac{1}{1}) + 0 + \frac{0}{lim_{x \to 0} x}$

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$2 (\frac{1}{1}) + 0 + \frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 (\frac{1}{1}) + 0 + \frac{0}{0}$

Since $\frac{0}{0}$ is of indeterminate form, apply L\'Hospital\'s Rule. L\'Hospital\'s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{\tan (x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [x]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$2 (\frac{1}{1}) + 0 + lim_{x \to 0} \frac{\frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [x]}$

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$2 (\frac{1}{1}) + 0 + lim_{x \to 0} \frac{\sec^{2} (x)}{\frac{d}{d x} [x]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 (\frac{1}{1}) + 0 + lim_{x \to 0} \frac{\sec^{2} (x)}{1}$

Divide $\sec^{2} (x)$ by $1$ .

$2 (\frac{1}{1}) + 0 + lim_{x \to 0} \sec^{2} (x)$

Step 5

Evaluate the limit.

Move the exponent $2$ from $\sec^{2} (x)$ outside the limit using the Limits Power Rule.

$2 (\frac{1}{1}) + 0 + {(lim_{x \to 0} \sec (x))}^{2}$

Move the limit inside the trig function because secant is continuous.

$2 (\frac{1}{1}) + 0 + \sec^{2} (lim_{x \to 0} x)$

Simplify terms.

Add $2 (\frac{1}{1})$ and $0$ .

$2 (\frac{1}{1}) + \sec^{2} (lim_{x \to 0} x)$

Simplify each term.

Cancel the common factor of $1$ .

Cancel the common factor.

$2 (\frac{1}{1}) + \sec^{2} (lim_{x \to 0} x)$

Rewrite the expression.

$2 \cdot 1 + \sec^{2} (lim_{x \to 0} x)$

Multiply $2$ by $1$ .

$2 + \sec^{2} (lim_{x \to 0} x)$

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$2 + \sec^{2} (0)$

The exact value of $\sec (0)$ is $1$ .

$2 + 1^{2}$

One to any power is one.

$2 + 1$

Add $2$ and $1$ .

$3$

Evaluate limit as x approaches 0 of 2/((sin(x))/x)+(1-cos(x))/x+(tan(x))/x

Related Questions

Support LCMGCF.com by disabling your adblocker.