Calculus Examples

$lim_{t \to 0} \frac{\sin (4 t)}{t \cos (t)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{t \to 0} \sin (4 t)}{lim_{t \to 0} t \cos (t)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{\sin (lim_{t \to 0} 4 t)}{lim_{t \to 0} t \cos (t)}$

Step 1.1.2.1.2

Move the term $4$ outside of the limit because it is constant with respect to $t$ .

$\frac{\sin (4 lim_{t \to 0} t)}{lim_{t \to 0} t \cos (t)}$

Step 1.1.2.2

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

$\frac{\sin (4 \cdot 0)}{lim_{t \to 0} t \cos (t)}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $4$ by $0$ .

$\frac{\sin (0)}{lim_{t \to 0} t \cos (t)}$

Step 1.1.2.3.2

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

$\frac{0}{lim_{t \to 0} t \cos (t)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{t \to 0} t \cdot lim_{t \to 0} \cos (t)}$

Step 1.1.3.2

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

$\frac{0}{lim_{t \to 0} t \cdot \cos (lim_{t \to 0} t)}$

Step 1.1.3.3

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

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Step 1.1.3.3.1

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

$\frac{0}{0 \cdot \cos (lim_{t \to 0} t)}$

Step 1.1.3.3.2

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

$\frac{0}{0 \cdot \cos (0)}$

Step 1.1.3.4

Simplify the answer.

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Step 1.1.3.4.1

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

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

Step 1.1.3.4.2

Multiply $0$ by $1$ .

$\frac{0}{0}$

Step 1.1.3.4.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.5

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{t \to 0} \frac{\sin (4 t)}{t \cos (t)} = lim_{t \to 0} \frac{\frac{d}{d t} [\sin (4 t)]}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{t \to 0} \frac{\frac{d}{d t} [\sin (4 t)]}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \sin (t)$ and $g (t) = 4 t$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $4 t$ .

$lim_{t \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d t} [4 t]}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.2.2

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

$lim_{t \to 0} \frac{\cos (u) \frac{d}{d t} [4 t]}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $4 t$ .

$lim_{t \to 0} \frac{\cos (4 t) \frac{d}{d t} [4 t]}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.3

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

$lim_{t \to 0} \frac{\cos (4 t) (4 \frac{d}{d t} [t])}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.4

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

$lim_{t \to 0} \frac{\cos (4 t) (4 \cdot 1)}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.5

Multiply $4$ by $1$ .

$lim_{t \to 0} \frac{\cos (4 t) \cdot 4}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.6

Move $4$ to the left of $\cos (4 t)$ .

$lim_{t \to 0} \frac{4 \cdot \cos (4 t)}{\frac{d}{d t} [t \cos (t)]}$

Step 1.3.7

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t$ and $g (t) = \cos (t)$ .

$lim_{t \to 0} \frac{4 \cos (4 t)}{t \frac{d}{d t} [\cos (t)] + \cos (t) \frac{d}{d t} [t]}$

Step 1.3.8

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

$lim_{t \to 0} \frac{4 \cos (4 t)}{t (- \sin (t)) + \cos (t) \frac{d}{d t} [t]}$

Step 1.3.9

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

$lim_{t \to 0} \frac{4 \cos (4 t)}{t (- \sin (t)) + \cos (t) \cdot 1}$

Step 1.3.10

Multiply $\cos (t)$ by $1$ .

$lim_{t \to 0} \frac{4 \cos (4 t)}{t (- \sin (t)) + \cos (t)}$

Step 1.3.11

Reorder terms.

$lim_{t \to 0} \frac{4 \cos (4 t)}{- t \sin (t) + \cos (t)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $4$ outside of the limit because it is constant with respect to $t$ .

$4 lim_{t \to 0} \frac{\cos (4 t)}{- t \sin (t) + \cos (t)}$

Step 2.2

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

$4 \frac{lim_{t \to 0} \cos (4 t)}{lim_{t \to 0} - t \sin (t) + \cos (t)}$

Step 2.3

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

$4 \frac{\cos (lim_{t \to 0} 4 t)}{lim_{t \to 0} - t \sin (t) + \cos (t)}$

Step 2.4

Move the term $4$ outside of the limit because it is constant with respect to $t$ .

$4 \frac{\cos (4 lim_{t \to 0} t)}{lim_{t \to 0} - t \sin (t) + \cos (t)}$

Step 2.5

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

$4 \frac{\cos (4 lim_{t \to 0} t)}{- lim_{t \to 0} t \sin (t) + lim_{t \to 0} \cos (t)}$

Step 2.6

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

$4 \frac{\cos (4 lim_{t \to 0} t)}{- lim_{t \to 0} t \cdot lim_{t \to 0} \sin (t) + lim_{t \to 0} \cos (t)}$

Step 2.7

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

$4 \frac{\cos (4 lim_{t \to 0} t)}{- lim_{t \to 0} t \cdot \sin (lim_{t \to 0} t) + lim_{t \to 0} \cos (t)}$

Step 2.8

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

$4 \frac{\cos (4 lim_{t \to 0} t)}{- lim_{t \to 0} t \cdot \sin (lim_{t \to 0} t) + \cos (lim_{t \to 0} t)}$

Step 3

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

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Step 3.1

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

$4 \frac{\cos (4 \cdot 0)}{- lim_{t \to 0} t \cdot \sin (lim_{t \to 0} t) + \cos (lim_{t \to 0} t)}$

Step 3.2

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

$4 \frac{\cos (4 \cdot 0)}{0 \cdot \sin (lim_{t \to 0} t) + \cos (lim_{t \to 0} t)}$

Step 3.3

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

$4 \frac{\cos (4 \cdot 0)}{0 \cdot \sin (0) + \cos (lim_{t \to 0} t)}$

Step 3.4

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

$4 \frac{\cos (4 \cdot 0)}{0 \cdot \sin (0) + \cos (0)}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $4$ by $0$ .

$4 \frac{\cos (0)}{0 \cdot \sin (0) + \cos (0)}$

Step 4.1.2

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

$4 \frac{1}{0 \cdot \sin (0) + \cos (0)}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

$4 \frac{1}{0 \cdot 0 + \cos (0)}$

Step 4.2.2

Multiply $0$ by $0$ .

$4 \frac{1}{0 + \cos (0)}$

Step 4.2.3

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

$4 \frac{1}{0 + 1}$

Step 4.2.4

Add $0$ and $1$ .

$4 (\frac{1}{1})$

Step 4.3

Cancel the common factor of $1$ .

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Step 4.3.1

Cancel the common factor.

$4 (\frac{1}{1})$

Step 4.3.2

Rewrite the expression.

$4 \cdot 1$

Step 4.4

Multiply $4$ by $1$ .

$4$