Simplify limit as h approaches 0 of ((x+h)^3-x^3)/h

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

Step 1

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.

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

Evaluate the limit of the numerator.

Evaluate the limit.

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

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

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

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

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

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

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

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

Evaluate the limit of $x^{3}$ which is constant as $h$ approaches $0$ .

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

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

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

Combine the opposite terms in ${(x + 0)}^{3} - x^{3}$ .

Add $x$ and $0$ .

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

Subtract $x^{3}$ from $x^{3}$ .

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

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.

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

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

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

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

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

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

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 $g (h) = x + h$ .

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

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

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

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

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

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

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

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

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

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

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

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

Add $0$ and $1$ .

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

Multiply $3$ by $1$ .

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

Since $- x^{3}$ is constant with respect to $h$ , the derivative of $- x^{3}$ with respect to $h$ is $0$ .

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

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

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

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

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

Divide $3 {(x + h)}^{2}$ by $1$ .

$lim_{h \to 0} 3 {(x + h)}^{2}$

Step 2

Evaluate the limit.

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

$3 lim_{h \to 0} {(x + h)}^{2}$

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

$3 {(lim_{h \to 0} x + h)}^{2}$

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

$3 {(lim_{h \to 0} x + lim_{h \to 0} h)}^{2}$

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

$3 {(x + lim_{h \to 0} h)}^{2}$

Step 3

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

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

Step 4

Add $x$ and $0$ .

$3 x^{2}$

Simplify limit as h approaches 0 of ((x+h)^3-x^3)/h

Related Questions

Support LCMGCF.com by disabling your adblocker.