Calculus Examples

$lim_{h \to 0} \frac{(6 (x + h) + 3) - (6 x + 3)}{h}$

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{lim_{h \to 0} (6 (x + h) + 3) - (6 x + 3)}{lim_{h \to 0} h}$

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

Evaluate the limit of $(6 (x + h) + 3) - (6 x + 3)$ by plugging in $0$ for $h$ .

$\frac{6 (x + 0) + 3 - (6 x + 3)}{lim_{h \to 0} h}$

Step 1.1.2.2

Add $x$ and $0$ .

$\frac{6 x + 3 - (6 x + 3)}{lim_{h \to 0} h}$

Step 1.1.2.3

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1

Apply the distributive property.

$\frac{6 x + 3 - (6 x) - 1 \cdot 3}{lim_{h \to 0} h}$

Step 1.1.2.3.2

Multiply $6$ by $- 1$ .

$\frac{6 x + 3 - 6 x - 1 \cdot 3}{lim_{h \to 0} h}$

Step 1.1.2.3.3

Multiply $- 1$ by $3$ .

$\frac{6 x + 3 - 6 x - 3}{lim_{h \to 0} h}$

Step 1.1.2.4

Combine the opposite terms in $6 x + 3 - 6 x - 3$ .

Tap for more steps...

Step 1.1.2.4.1

Subtract $6 x$ from $6 x$ .

$\frac{0 + 3 - 3}{lim_{h \to 0} h}$

Step 1.1.2.4.2

Add $0$ and $3$ .

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

Step 1.1.2.4.3

Subtract $3$ from $3$ .

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

Step 1.1.3

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

$\frac{0}{0}$

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.

$lim_{h \to 0} \frac{(6 (x + h) + 3) - (6 x + 3)}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [(6 (x + h) + 3) - (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [(6 (x + h) + 3) - (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.2

By the Sum Rule, the derivative of $(6 (x + h) + 3) - (6 x + 3)$ with respect to $h$ is $\frac{d}{d h} [6 (x + h)] + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [6 (x + h)] + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.3

Evaluate $\frac{d}{d h} [6 (x + h)]$ .

Tap for more steps...

Step 1.3.3.1

Since $6$ is constant with respect to $h$ , the derivative of $6 (x + h)$ with respect to $h$ is $6 \frac{d}{d h} [x + h]$ .

$lim_{h \to 0} \frac{6 \frac{d}{d h} [x + h] + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.3.2

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

$lim_{h \to 0} \frac{6 (\frac{d}{d h} [x] + \frac{d}{d h} [h]) + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.3.3

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

$lim_{h \to 0} \frac{6 (0 + \frac{d}{d h} [h]) + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.3.4

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

$lim_{h \to 0} \frac{6 (0 + 1) + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.3.5

Add $0$ and $1$ .

$lim_{h \to 0} \frac{6 \cdot 1 + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.3.6

Multiply $6$ by $1$ .

$lim_{h \to 0} \frac{6 + \frac{d}{d h} [3] + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.4

Since $3$ is constant with respect to $h$ , the derivative of $3$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{6 + 0 + \frac{d}{d h} [- (6 x + 3)]}{\frac{d}{d h} [h]}$

Step 1.3.5

Since $- (6 x + 3)$ is constant with respect to $h$ , the derivative of $- (6 x + 3)$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{6 + 0 + 0}{\frac{d}{d h} [h]}$

Step 1.3.6

Combine terms.

Tap for more steps...

Step 1.3.6.1

Add $6$ and $0$ .

$lim_{h \to 0} \frac{6 + 0}{\frac{d}{d h} [h]}$

Step 1.3.6.2

Add $6$ and $0$ .

$lim_{h \to 0} \frac{6}{\frac{d}{d h} [h]}$

Step 1.3.7

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

$lim_{h \to 0} \frac{6}{1}$

Step 1.4

Divide $6$ by $1$ .

$lim_{h \to 0} 6$

Step 2

Evaluate the limit of $6$ which is constant as $h$ approaches $0$ .

$6$