Calculus Examples

$lim_{c \to 0} \frac{mg}{c} (1 - e^{- \frac{c t}{m}})$

Step 1

Evaluate the limit.

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Step 1.1

Multiply $\frac{mg}{c}$ by $1 - e^{- \frac{c t}{m}}$ .

$lim_{c \to 0} \frac{mg (1 - e^{- \frac{c t}{m}})}{c}$

Step 1.2

Move the term $mg$ outside of the limit because it is constant with respect to $c$ .

$mg lim_{c \to 0} \frac{1 - e^{- \frac{c t}{m}}}{c}$

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.

$mg \frac{lim_{c \to 0} 1 - e^{- \frac{c t}{m}}}{lim_{c \to 0} c}$

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

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

$mg \frac{lim_{c \to 0} 1 - lim_{c \to 0} e^{- \frac{c t}{m}}}{lim_{c \to 0} c}$

Step 2.1.2.1.2

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

$mg \frac{1 - lim_{c \to 0} e^{- \frac{c t}{m}}}{lim_{c \to 0} c}$

Step 2.1.2.1.3

Move the limit into the exponent.

$mg \frac{1 - e^{lim_{c \to 0} - \frac{c t}{m}}}{lim_{c \to 0} c}$

Step 2.1.2.1.4

Move the term $- 1$ outside of the limit because it is constant with respect to $c$ .

$mg \frac{1 - e^{- lim_{c \to 0} \frac{c t}{m}}}{lim_{c \to 0} c}$

Step 2.1.2.1.5

Move the term $\frac{t}{m}$ outside of the limit because it is constant with respect to $c$ .

$mg \frac{1 - e^{- \frac{t}{m} lim_{c \to 0} c}}{lim_{c \to 0} c}$

Step 2.1.2.2

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

$mg \frac{1 - e^{- \frac{t}{m} \cdot 0}}{lim_{c \to 0} c}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $- \frac{t}{m} \cdot 0$ .

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Step 2.1.2.3.1.1.1

Multiply $0$ by $- 1$ .

$mg \frac{1 - e^{0 \frac{t}{m}}}{lim_{c \to 0} c}$

Step 2.1.2.3.1.1.2

Multiply $0$ by $\frac{t}{m}$ .

$mg \frac{1 - e^{0}}{lim_{c \to 0} c}$

Step 2.1.2.3.1.2

Anything raised to $0$ is $1$ .

$mg \frac{1 - 1 \cdot 1}{lim_{c \to 0} c}$

Step 2.1.2.3.1.3

Multiply $- 1$ by $1$ .

$mg \frac{1 - 1}{lim_{c \to 0} c}$

Step 2.1.2.3.2

Subtract $1$ from $1$ .

$mg \frac{0}{lim_{c \to 0} c}$

Step 2.1.3

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

$mg \frac{0}{0}$

Step 2.1.4

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

Undefined

$mg \frac{0}{0}$

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.

$lim_{c \to 0} \frac{1 - e^{- \frac{c t}{m}}}{c} = lim_{c \to 0} \frac{\frac{d}{d c} [1 - e^{- \frac{c t}{m}}]}{\frac{d}{d c} [c]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$mg lim_{c \to 0} \frac{\frac{d}{d c} [1 - e^{- \frac{c t}{m}}]}{\frac{d}{d c} [c]}$

Step 2.3.2

By the Sum Rule, the derivative of $1 - e^{- \frac{c t}{m}}$ with respect to $c$ is $\frac{d}{d c} [1] + \frac{d}{d c} [- e^{- \frac{c t}{m}}]$ .

$mg lim_{c \to 0} \frac{\frac{d}{d c} [1] + \frac{d}{d c} [- e^{- \frac{c t}{m}}]}{\frac{d}{d c} [c]}$

Step 2.3.3

Since $1$ is constant with respect to $c$ , the derivative of $1$ with respect to $c$ is $0$ .

$mg lim_{c \to 0} \frac{0 + \frac{d}{d c} [- e^{- \frac{c t}{m}}]}{\frac{d}{d c} [c]}$

Step 2.3.4

Evaluate $\frac{d}{d c} [- e^{- \frac{c t}{m}}]$ .

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Step 2.3.4.1

Since $- 1$ is constant with respect to $c$ , the derivative of $- e^{- \frac{c t}{m}}$ with respect to $c$ is $- \frac{d}{d c} [e^{- \frac{c t}{m}}]$ .

$mg lim_{c \to 0} \frac{0 - \frac{d}{d c} [e^{- \frac{c t}{m}}]}{\frac{d}{d c} [c]}$

Step 2.3.4.2

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

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Step 2.3.4.2.1

To apply the Chain Rule, set $u$ as $- \frac{c t}{m}$ .

$mg lim_{c \to 0} \frac{0 - \frac{d}{d u} [e^{u}] \frac{d}{d c} [- \frac{c t}{m}]}{\frac{d}{d c} [c]}$

Step 2.3.4.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$mg lim_{c \to 0} \frac{0 - e^{u} \frac{d}{d c} [- \frac{c t}{m}]}{\frac{d}{d c} [c]}$

Step 2.3.4.2.3

Replace all occurrences of $u$ with $- \frac{c t}{m}$ .

$mg lim_{c \to 0} \frac{0 - e^{- \frac{c t}{m}} \frac{d}{d c} [- \frac{c t}{m}]}{\frac{d}{d c} [c]}$

Step 2.3.4.3

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

$mg lim_{c \to 0} \frac{0 - e^{- \frac{c t}{m}} \cdot - 1 \frac{t}{m} \frac{d}{d c} [c]}{\frac{d}{d c} [c]}$

Step 2.3.4.4

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

$mg lim_{c \to 0} \frac{0 - e^{- \frac{c t}{m}} \cdot - 1 \frac{t}{m} \cdot 1}{\frac{d}{d c} [c]}$

Step 2.3.4.5

Multiply $- 1$ by $1$ .

$mg lim_{c \to 0} \frac{0 - e^{- \frac{c t}{m}} \cdot - 1 \frac{t}{m}}{\frac{d}{d c} [c]}$

Step 2.3.4.6

Combine $e^{- \frac{c t}{m}}$ and $\frac{t}{m}$ .

$mg lim_{c \to 0} \frac{0 - - 1 \frac{e^{- \frac{c t}{m}} t}{m}}{\frac{d}{d c} [c]}$

Step 2.3.4.7

Multiply $- 1$ by $- 1$ .

$mg lim_{c \to 0} \frac{0 + 1 \frac{e^{- \frac{c t}{m}} t}{m}}{\frac{d}{d c} [c]}$

Step 2.3.4.8

Multiply $\frac{e^{- \frac{c t}{m}} t}{m}$ by $1$ .

$mg lim_{c \to 0} \frac{0 + \frac{e^{- \frac{c t}{m}} t}{m}}{\frac{d}{d c} [c]}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Add $0$ and $\frac{e^{- \frac{c t}{m}} t}{m}$ .

$mg lim_{c \to 0} \frac{\frac{e^{- \frac{c t}{m}} t}{m}}{\frac{d}{d c} [c]}$

Step 2.3.5.2

Reorder factors in $\frac{e^{- \frac{c t}{m}} t}{m}$ .

$mg lim_{c \to 0} \frac{\frac{t e^{- \frac{c t}{m}}}{m}}{\frac{d}{d c} [c]}$

Step 2.3.6

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

$mg lim_{c \to 0} \frac{\frac{t e^{- \frac{c t}{m}}}{m}}{1}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$mg lim_{c \to 0} \frac{t e^{- \frac{c t}{m}}}{m} \cdot 1$

Step 2.5

Multiply $\frac{t e^{- \frac{c t}{m}}}{m}$ by $1$ .

$mg lim_{c \to 0} \frac{t e^{- \frac{c t}{m}}}{m}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $\frac{t}{m}$ outside of the limit because it is constant with respect to $c$ .

$mg \frac{t}{m} lim_{c \to 0} e^{- \frac{c t}{m}}$

Step 3.2

Move the limit into the exponent.

$mg \frac{t}{m} e^{lim_{c \to 0} - \frac{c t}{m}}$

Step 3.3

Move the term $- 1$ outside of the limit because it is constant with respect to $c$ .

$mg \frac{t}{m} e^{- lim_{c \to 0} \frac{c t}{m}}$

Step 3.4

Move the term $\frac{t}{m}$ outside of the limit because it is constant with respect to $c$ .

$mg \frac{t}{m} e^{- \frac{t}{m} lim_{c \to 0} c}$

Step 4

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

$mg \frac{t}{m} e^{- \frac{t}{m} \cdot 0}$

Step 5

Simplify the answer.

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Step 5.1

Combine $mg$ and $\frac{t}{m}$ .

$\frac{t}{m} mg e^{- \frac{t}{m} \cdot 0}$

Step 5.2

Multiply $- \frac{t}{m} \cdot 0$ .

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Step 5.2.1

Multiply $0$ by $- 1$ .

$\frac{t}{m} mg e^{0 \frac{t}{m}}$

Step 5.2.2

Multiply $0$ by $\frac{t}{m}$ .

$\frac{t}{m} mg e^{0}$

Step 5.3

Anything raised to $0$ is $1$ .

$\frac{t}{m} mg \cdot 1$

Step 5.4

Multiply $\frac{t}{m}$ by $1$ .

$\frac{t}{m} mg$