Calculus Examples

$lim_{x \to 0} \frac{\sin (A x^{2})}{B x}$

Step 1

Move the term $\frac{1}{B}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{B} lim_{x \to 0} \frac{\sin (A x^{2})}{x}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{1}{B} \cdot \frac{lim_{x \to 0} \sin (A x^{2})}{lim_{x \to 0} x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

$\frac{1}{B} \cdot \frac{\sin (lim_{x \to 0} A x^{2})}{lim_{x \to 0} x}$

Step 2.1.2.1.2

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

$\frac{1}{B} \cdot \frac{\sin (A lim_{x \to 0} x^{2})}{lim_{x \to 0} x}$

Step 2.1.2.1.3

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

$\frac{1}{B} \cdot \frac{\sin (A {(lim_{x \to 0} x)}^{2})}{lim_{x \to 0} x}$

Step 2.1.2.2

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

$\frac{1}{B} \cdot \frac{\sin (A \cdot 0^{2})}{lim_{x \to 0} x}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{B} \cdot \frac{\sin (A \cdot 0)}{lim_{x \to 0} x}$

Step 2.1.2.3.2

Multiply $A$ by $0$ .

$\frac{1}{B} \cdot \frac{\sin (0)}{lim_{x \to 0} x}$

Step 2.1.2.3.3

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

$\frac{1}{B} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

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

$\frac{1}{B} \cdot \frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{1}{B} \cdot \frac{0}{0}$

Step 2.2

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.

$lim_{x \to 0} \frac{\sin (A x^{2})}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (A x^{2})]}{\frac{d}{d x} [x]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{B} lim_{x \to 0} \frac{\frac{d}{d x} [\sin (A x^{2})]}{\frac{d}{d x} [x]}$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = A x^{2}$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $A x^{2}$ .

$\frac{1}{B} lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [A x^{2}]}{\frac{d}{d x} [x]}$

Step 2.3.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\frac{1}{B} lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [A x^{2}]}{\frac{d}{d x} [x]}$

Step 2.3.2.3

Replace all occurrences of $u$ with $A x^{2}$ .

$\frac{1}{B} lim_{x \to 0} \frac{\cos (A x^{2}) \frac{d}{d x} [A x^{2}]}{\frac{d}{d x} [x]}$

Step 2.3.3

Since $A$ is constant with respect to $x$ , the derivative of $A x^{2}$ with respect to $x$ is $A \frac{d}{d x} [x^{2}]$ .

$\frac{1}{B} lim_{x \to 0} \frac{\cos (A x^{2}) A \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [x]}$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{B} lim_{x \to 0} \frac{\cos (A x^{2}) A \cdot 2 x}{\frac{d}{d x} [x]}$

Step 2.3.5

Move $2$ to the left of $A$ .

$\frac{1}{B} lim_{x \to 0} \frac{\cos (A x^{2}) \cdot 2 \cdot A x}{\frac{d}{d x} [x]}$

Step 2.3.6

Reorder the factors of $\cos (A x^{2}) (2 A x)$ .

$\frac{1}{B} lim_{x \to 0} \frac{2 A x \cos (A x^{2})}{\frac{d}{d x} [x]}$

Step 2.3.7

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

$\frac{1}{B} lim_{x \to 0} \frac{2 A x \cos (A x^{2})}{1}$

Step 2.4

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

$\frac{1}{B} lim_{x \to 0} 2 A x \cos (A x^{2})$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{1}{B} \cdot 2 A lim_{x \to 0} x \cos (A x^{2})$

Step 3.2

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

$\frac{1}{B} \cdot 2 A lim_{x \to 0} x \cdot lim_{x \to 0} \cos (A x^{2})$

Step 3.3

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

$\frac{1}{B} \cdot 2 A lim_{x \to 0} x \cdot \cos (lim_{x \to 0} A x^{2})$

Step 3.4

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

$\frac{1}{B} \cdot 2 A lim_{x \to 0} x \cdot \cos (A lim_{x \to 0} x^{2})$

Step 3.5

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

$\frac{1}{B} \cdot 2 A lim_{x \to 0} x \cdot \cos (A {(lim_{x \to 0} x)}^{2})$

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

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

$\frac{1}{B} \cdot 2 A \cdot 0 \cdot \cos (A {(lim_{x \to 0} x)}^{2})$

Step 4.2

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

$\frac{1}{B} \cdot 2 A \cdot 0 \cdot \cos (A \cdot 0^{2})$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{B}$ and $2$ .

$\frac{2}{B} A \cdot 0 \cdot \cos (A \cdot 0^{2})$

Step 5.2

Combine $\frac{2}{B}$ and $A$ .

$\frac{2 A}{B} \cdot 0 \cdot \cos (A \cdot 0^{2})$

Step 5.3

Multiply $\frac{2 A}{B}$ by $0$ .

$0 \cdot \cos (A \cdot 0^{2})$

Step 5.4

Raising $0$ to any positive power yields $0$ .

$0 \cdot \cos (A \cdot 0)$

Step 5.5

Multiply $A$ by $0$ .

$0 \cdot \cos (0)$

Step 5.6

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

$0 \cdot 1$

Step 5.7

Multiply $0$ by $1$ .

$0$