Calculus Examples

$lim_{x \to 3} \frac{{(x - 2)}^{2} - {(x - 4)}^{2}}{x - 3}$

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.

$\frac{lim_{x \to 3} {(x - 2)}^{2} - {(x - 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} {(x - 2)}^{2} - lim_{x \to 3} {(x - 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.2

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

$\frac{{(lim_{x \to 3} x - 2)}^{2} - lim_{x \to 3} {(x - 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x - lim_{x \to 3} 2)}^{2} - lim_{x \to 3} {(x - 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.4

Evaluate the limit of $2$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x - 1 \cdot 2)}^{2} - lim_{x \to 3} {(x - 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.5

Move the exponent $2$ from ${(x - 4)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 3} x - 1 \cdot 2)}^{2} - {(lim_{x \to 3} x - 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x - 1 \cdot 2)}^{2} - {(lim_{x \to 3} x - lim_{x \to 3} 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.7

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x - 1 \cdot 2)}^{2} - {(lim_{x \to 3} x - 1 \cdot 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.8

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 1.1.2.8.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{{(3 - 1 \cdot 2)}^{2} - {(lim_{x \to 3} x - 1 \cdot 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.8.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{{(3 - 1 \cdot 2)}^{2} - {(3 - 1 \cdot 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.9

Simplify the answer.

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Step 1.1.2.9.1

Simplify each term.

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Step 1.1.2.9.1.1

Multiply $- 1$ by $2$ .

$\frac{{(3 - 2)}^{2} - {(3 - 1 \cdot 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.9.1.2

Subtract $2$ from $3$ .

$\frac{1^{2} - {(3 - 1 \cdot 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.9.1.3

One to any power is one.

$\frac{1 - {(3 - 1 \cdot 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.9.1.4

Multiply $- 1$ by $4$ .

$\frac{1 - {(3 - 4)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.9.1.5

Subtract $4$ from $3$ .

$\frac{1 - {(- 1)}^{2}}{lim_{x \to 3} x - 3}$

Step 1.1.2.9.1.6

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.1.2.9.1.6.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 1.1.2.9.1.6.1.1

Raise $- 1$ to the power of $1$ .

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

Step 1.1.2.9.1.6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.2.9.1.6.2

Add $1$ and $2$ .

$\frac{1 + {(- 1)}^{3}}{lim_{x \to 3} x - 3}$

Step 1.1.2.9.1.7

Raise $- 1$ to the power of $3$ .

$\frac{1 - 1}{lim_{x \to 3} x - 3}$

Step 1.1.2.9.2

Subtract $1$ from $1$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 1.1.3.1.2

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{0}{3 - 1 \cdot 3}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $3$ .

$\frac{0}{3 - 3}$

Step 1.1.3.3.2

Subtract $3$ from $3$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 3} \frac{{(x - 2)}^{2} - {(x - 4)}^{2}}{x - 3} = lim_{x \to 3} \frac{\frac{d}{d x} [{(x - 2)}^{2} - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [{(x - 2)}^{2} - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.2

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [(x - 2) (x - 2) - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.3

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.3.3.1

Apply the distributive property.

$lim_{x \to 3} \frac{\frac{d}{d x} [x (x - 2) - 2 (x - 2) - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.3.2

Apply the distributive property.

$lim_{x \to 3} \frac{\frac{d}{d x} [x \cdot x + x \cdot - 2 - 2 (x - 2) - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.3.3

Apply the distributive property.

$lim_{x \to 3} \frac{\frac{d}{d x} [x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $x$ by $x$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.4.1.2

Move $- 2$ to the left of $x$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.4.1.3

Multiply $- 2$ by $- 2$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 2 x - 2 x + 4 - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.4.2

Subtract $2 x$ from $- 2 x$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - {(x - 4)}^{2}]}{\frac{d}{d x} [x - 3]}$

Step 1.3.5

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - ((x - 4) (x - 4))]}{\frac{d}{d x} [x - 3]}$

Step 1.3.6

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 1.3.6.1

Apply the distributive property.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - (x (x - 4) - 4 (x - 4))]}{\frac{d}{d x} [x - 3]}$

Step 1.3.6.2

Apply the distributive property.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - (x \cdot x + x \cdot - 4 - 4 (x - 4))]}{\frac{d}{d x} [x - 3]}$

Step 1.3.6.3

Apply the distributive property.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.7

Simplify and combine like terms.

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Step 1.3.7.1

Simplify each term.

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Step 1.3.7.1.1

Multiply $x$ by $x$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.7.1.2

Move $- 4$ to the left of $x$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.7.1.3

Multiply $- 4$ by $- 4$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - (x^{2} - 4 x - 4 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.7.2

Subtract $4 x$ from $- 4 x$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 4 - (x^{2} - 8 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.8

By the Sum Rule, the derivative of $x^{2} - 4 x + 4 - (x^{2} - 8 x + 16)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4] + \frac{d}{d x} [- (x^{2} - 8 x + 16)]$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4] + \frac{d}{d x} [- (x^{2} - 8 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 3} \frac{2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4] + \frac{d}{d x} [- (x^{2} - 8 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.10

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 1.3.10.1

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

$lim_{x \to 3} \frac{2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [4] + \frac{d}{d x} [- (x^{2} - 8 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.10.2

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

$lim_{x \to 3} \frac{2 x - 4 \cdot 1 + \frac{d}{d x} [4] + \frac{d}{d x} [- (x^{2} - 8 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.10.3

Multiply $- 4$ by $1$ .

$lim_{x \to 3} \frac{2 x - 4 + \frac{d}{d x} [4] + \frac{d}{d x} [- (x^{2} - 8 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.11

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 + \frac{d}{d x} [- (x^{2} - 8 x + 16)]}{\frac{d}{d x} [x - 3]}$

Step 1.3.12

Evaluate $\frac{d}{d x} [- (x^{2} - 8 x + 16)]$ .

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Step 1.3.12.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x^{2} - 8 x + 16)$ with respect to $x$ is $- \frac{d}{d x} [x^{2} - 8 x + 16]$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 - \frac{d}{d x} [x^{2} - 8 x + 16]}{\frac{d}{d x} [x - 3]}$

Step 1.3.12.2

By the Sum Rule, the derivative of $x^{2} - 8 x + 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [16]$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 - (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [16])}{\frac{d}{d x} [x - 3]}$

Step 1.3.12.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 - (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [16])}{\frac{d}{d x} [x - 3]}$

Step 1.3.12.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 - (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [16])}{\frac{d}{d x} [x - 3]}$

Step 1.3.12.5

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

$lim_{x \to 3} \frac{2 x - 4 + 0 - (2 x - 8 \cdot 1 + \frac{d}{d x} [16])}{\frac{d}{d x} [x - 3]}$

Step 1.3.12.6

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 - (2 x - 8 \cdot 1 + 0)}{\frac{d}{d x} [x - 3]}$

Step 1.3.12.7

Multiply $- 8$ by $1$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 - (2 x - 8 + 0)}{\frac{d}{d x} [x - 3]}$

Step 1.3.12.8

Add $2 x - 8$ and $0$ .

$lim_{x \to 3} \frac{2 x - 4 + 0 - (2 x - 8)}{\frac{d}{d x} [x - 3]}$

Step 1.3.13

Simplify.

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Step 1.3.13.1

Apply the distributive property.

$lim_{x \to 3} \frac{2 x - 4 + 0 - (2 x) - - 8}{\frac{d}{d x} [x - 3]}$

Step 1.3.13.2

Combine terms.

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Step 1.3.13.2.1

Add $2 x - 4$ and $0$ .

$lim_{x \to 3} \frac{2 x - 4 - (2 x) - - 8}{\frac{d}{d x} [x - 3]}$

Step 1.3.13.2.2

Multiply $2$ by $- 1$ .

$lim_{x \to 3} \frac{2 x - 4 - 2 x - - 8}{\frac{d}{d x} [x - 3]}$

Step 1.3.13.2.3

Multiply $- 1$ by $- 8$ .

$lim_{x \to 3} \frac{2 x - 4 - 2 x + 8}{\frac{d}{d x} [x - 3]}$

Step 1.3.13.2.4

Subtract $2 x$ from $2 x$ .

$lim_{x \to 3} \frac{0 - 4 + 8}{\frac{d}{d x} [x - 3]}$

Step 1.3.13.2.5

Subtract $4$ from $0$ .

$lim_{x \to 3} \frac{- 4 + 8}{\frac{d}{d x} [x - 3]}$

Step 1.3.13.2.6

Add $- 4$ and $8$ .

$lim_{x \to 3} \frac{4}{\frac{d}{d x} [x - 3]}$

Step 1.3.14

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

$lim_{x \to 3} \frac{4}{\frac{d}{d x} [x] + \frac{d}{d x} [- 3]}$

Step 1.3.15

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

$lim_{x \to 3} \frac{4}{1 + \frac{d}{d x} [- 3]}$

Step 1.3.16

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

$lim_{x \to 3} \frac{4}{1 + 0}$

Step 1.3.17

Add $1$ and $0$ .

$lim_{x \to 3} \frac{4}{1}$

Step 1.4

Divide $4$ by $1$ .

$lim_{x \to 3} 4$

Step 2

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

$4$