Evaluate limit as h approaches 0 of ((7+h)^3-343)/h

$lim_{h \to 0} \frac{{(7 + h)}^{3} - 343}{h}$

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

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

$\frac{lim_{h \to 0} {(7 + h)}^{3} - 343}{lim_{h \to 0} h}$

Evaluate the limit of the numerator.

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Take the limit of each term.

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

$\frac{lim_{h \to 0} {(7 + h)}^{3} - lim_{h \to 0} 343}{lim_{h \to 0} h}$

Move the exponent $3$ from ${(7 + h)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{h \to 0} 7 + h)}^{3} - lim_{h \to 0} 343}{lim_{h \to 0} h}$

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

$\frac{{(lim_{h \to 0} 7 + lim_{h \to 0} h)}^{3} - lim_{h \to 0} 343}{lim_{h \to 0} h}$

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

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Evaluate the limit of $7$ which is constant as $h$ approaches $0$ .

$\frac{{(7 + lim_{h \to 0} h)}^{3} - lim_{h \to 0} 343}{lim_{h \to 0} h}$

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

$\frac{{(7 + 0)}^{3} - lim_{h \to 0} 343}{lim_{h \to 0} h}$

Evaluate the limit of $343$ which is constant as $h$ approaches $0$ .

$\frac{{(7 + 0)}^{3} - 343}{lim_{h \to 0} h}$

Simplify the answer.

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Simplify each term.

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Add $7$ and $0$ .

$\frac{7^{3} - 343}{lim_{h \to 0} h}$

Raise $7$ to the power of $3$ .

$\frac{343 - 343}{lim_{h \to 0} h}$

Subtract $343$ from $343$ .

$\frac{0}{lim_{h \to 0} h}$

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

$\frac{0}{0}$

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

Undefined

$\frac{0}{0}$

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.

$lim_{h \to 0} \frac{{(7 + h)}^{3} - 343}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [{(7 + h)}^{3} - 343]}{\frac{d}{d h} [h]}$

Find the derivative of the numerator and denominator.

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Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [{(7 + h)}^{3} - 343]}{\frac{d}{d h} [h]}$

By the Sum Rule, the derivative of ${(7 + h)}^{3} - 343$ with respect to $h$ is $\frac{d}{d h} [{(7 + h)}^{3}] + \frac{d}{d h} [- 343]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [{(7 + h)}^{3}] + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

Evaluate $\frac{d}{d h} [{(7 + h)}^{3}]$ .

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Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{3}$ and $g (h) = 7 + h$ .

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To apply the Chain Rule, set $u$ as $7 + h$ .

$lim_{h \to 0} \frac{\frac{d}{d u} [u^{3}] \frac{d}{d h} [7 + h] + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$lim_{h \to 0} \frac{3 u^{2} \frac{d}{d h} [7 + h] + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

Replace all occurrences of $u$ with $7 + h$ .

$lim_{h \to 0} \frac{3 {(7 + h)}^{2} \frac{d}{d h} [7 + h] + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

By the Sum Rule, the derivative of $7 + h$ with respect to $h$ is $\frac{d}{d h} [7] + \frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{3 {(7 + h)}^{2} (\frac{d}{d h} [7] + \frac{d}{d h} [h]) + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

Since $7$ is constant with respect to $h$ , the derivative of $7$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{3 {(7 + h)}^{2} (0 + \frac{d}{d h} [h]) + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

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

$lim_{h \to 0} \frac{3 {(7 + h)}^{2} (0 + 1) + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

Add $0$ and $1$ .

$lim_{h \to 0} \frac{3 {(7 + h)}^{2} \cdot 1 + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

Multiply $3$ by $1$ .

$lim_{h \to 0} \frac{3 {(7 + h)}^{2} + \frac{d}{d h} [- 343]}{\frac{d}{d h} [h]}$

Since $- 343$ is constant with respect to $h$ , the derivative of $- 343$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{3 {(7 + h)}^{2} + 0}{\frac{d}{d h} [h]}$

Add $3 {(7 + h)}^{2}$ and $0$ .

$lim_{h \to 0} \frac{3 {(7 + h)}^{2}}{\frac{d}{d h} [h]}$

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

$lim_{h \to 0} \frac{3 {(7 + h)}^{2}}{1}$

Take the limit of each term.

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

$\frac{(lim_{h \to 0} 3 {(7 + h)}^{2})}{(lim_{h \to 0} 1)}$

Move the term $3$ outside of the limit because it is constant with respect to $h$ .

$\frac{3 lim_{h \to 0} {(7 + h)}^{2}}{lim_{h \to 0} 1}$

Move the exponent $2$ from ${(7 + h)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{h \to 0} 7 + h)}^{2}}{lim_{h \to 0} 1}$

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

$\frac{3 {(lim_{h \to 0} 7 + lim_{h \to 0} h)}^{2}}{lim_{h \to 0} 1}$

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

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Evaluate the limit of $7$ which is constant as $h$ approaches $0$ .

$\frac{3 {(7 + lim_{h \to 0} h)}^{2}}{lim_{h \to 0} 1}$

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

$\frac{3 {(7 + 0)}^{2}}{lim_{h \to 0} 1}$

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

$\frac{3 {(7 + 0)}^{2}}{1}$

Simplify the answer.

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Divide $3 {(7 + 0)}^{2}$ by $1$ .

$3 {(7 + 0)}^{2}$

Add $7$ and $0$ .

$3 \cdot 7^{2}$

Raise $7$ to the power of $2$ .

$3 \cdot 49$

Multiply $3$ by $49$ .

$147$

Evaluate limit as h approaches 0 of ((7+h)^3-343)/h

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