Calculus Examples

$lim_{x \to 3} \frac{x^{2} - 4 x + 3}{x^{2} - 8 x + 15}$

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.

$\frac{lim_{x \to 3} x^{2} - 4 x + 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

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 $3$ .

$\frac{lim_{x \to 3} x^{2} - lim_{x \to 3} 4 x + lim_{x \to 3} 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.2

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

$\frac{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 4 x + lim_{x \to 3} 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 4 lim_{x \to 3} x + lim_{x \to 3} 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.4

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

$\frac{{(lim_{x \to 3} x)}^{2} - 4 lim_{x \to 3} x + 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

$\frac{3^{2} - 4 lim_{x \to 3} x + 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.5.2

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

$\frac{3^{2} - 4 \cdot 3 + 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

Raise $3$ to the power of $2$ .

$\frac{9 - 4 \cdot 3 + 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.6.1.2

Multiply $- 4$ by $3$ .

$\frac{9 - 12 + 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.6.2

Subtract $12$ from $9$ .

$\frac{- 3 + 3}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.2.6.3

Add $- 3$ and $3$ .

$\frac{0}{lim_{x \to 3} x^{2} - 8 x + 15}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 3} x^{2} - lim_{x \to 3} 8 x + lim_{x \to 3} 15}$

Step 1.1.3.2

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

$\frac{0}{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 8 x + lim_{x \to 3} 15}$

Step 1.1.3.3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{{(lim_{x \to 3} x)}^{2} - 8 lim_{x \to 3} x + lim_{x \to 3} 15}$

Step 1.1.3.4

Evaluate the limit of $15$ which is constant as $x$ approaches $3$ .

$\frac{0}{{(lim_{x \to 3} x)}^{2} - 8 lim_{x \to 3} x + 15}$

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

$\frac{0}{3^{2} - 8 lim_{x \to 3} x + 15}$

Step 1.1.3.5.2

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

$\frac{0}{3^{2} - 8 \cdot 3 + 15}$

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

Raise $3$ to the power of $2$ .

$\frac{0}{9 - 8 \cdot 3 + 15}$

Step 1.1.3.6.1.2

Multiply $- 8$ by $3$ .

$\frac{0}{9 - 24 + 15}$

Step 1.1.3.6.2

Subtract $24$ from $9$ .

$\frac{0}{- 15 + 15}$

Step 1.1.3.6.3

Add $- 15$ and $15$ .

$\frac{0}{0}$

Step 1.1.3.6.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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.

$lim_{x \to 3} \frac{x^{2} - 4 x + 3}{x^{2} - 8 x + 15} = lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 3]}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 4 x + 3]}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.2

By the Sum Rule, the derivative of $x^{2} - 4 x + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.3

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

$lim_{x \to 3} \frac{2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [3]}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 3} \frac{2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [3]}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.4.2

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

$lim_{x \to 3} \frac{2 x - 4 \cdot 1 + \frac{d}{d x} [3]}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.4.3

Multiply $- 4$ by $1$ .

$lim_{x \to 3} \frac{2 x - 4 + \frac{d}{d x} [3]}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.5

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

$lim_{x \to 3} \frac{2 x - 4 + 0}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.6

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

$lim_{x \to 3} \frac{2 x - 4}{\frac{d}{d x} [x^{2} - 8 x + 15]}$

Step 1.3.7

By the Sum Rule, the derivative of $x^{2} - 8 x + 15$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [15]$ .

$lim_{x \to 3} \frac{2 x - 4}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [15]}$

Step 1.3.8

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

$lim_{x \to 3} \frac{2 x - 4}{2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [15]}$

Step 1.3.9

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 1.3.9.1

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

$lim_{x \to 3} \frac{2 x - 4}{2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [15]}$

Step 1.3.9.2

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

$lim_{x \to 3} \frac{2 x - 4}{2 x - 8 \cdot 1 + \frac{d}{d x} [15]}$

Step 1.3.9.3

Multiply $- 8$ by $1$ .

$lim_{x \to 3} \frac{2 x - 4}{2 x - 8 + \frac{d}{d x} [15]}$

Step 1.3.10

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{2 x - 4}{2 x - 8 + 0}$

Step 1.3.11

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

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

Step 1.4

Cancel the common factor of $2 x - 4$ and $2 x - 8$ .

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Step 1.4.1

Factor $2$ out of $2 x$ .

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

Step 1.4.2

Factor $2$ out of $- 4$ .

$lim_{x \to 3} \frac{2 (x) + 2 \cdot - 2}{2 x - 8}$

Step 1.4.3

Factor $2$ out of $2 (x) + 2 (- 2)$ .

$lim_{x \to 3} \frac{2 (x - 2)}{2 x - 8}$

Step 1.4.4

Cancel the common factors.

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Step 1.4.4.1

Factor $2$ out of $2 x$ .

$lim_{x \to 3} \frac{2 (x - 2)}{2 (x) - 8}$

Step 1.4.4.2

Factor $2$ out of $- 8$ .

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

Step 1.4.4.3

Factor $2$ out of $2 x + 2 \cdot - 4$ .

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

Step 1.4.4.4

Cancel the common factor.

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

Step 1.4.4.5

Rewrite the expression.

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

$\frac{3 - 1 \cdot 2}{3 - 1 \cdot 4}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $2$ .

$\frac{3 - 2}{3 - 1 \cdot 4}$

Step 4.1.2

Subtract $2$ from $3$ .

$\frac{1}{3 - 1 \cdot 4}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $- 1$ by $4$ .

$\frac{1}{3 - 4}$

Step 4.2.2

Subtract $4$ from $3$ .

$\frac{1}{- 1}$

Step 4.3

Divide $1$ by $- 1$ .

$- 1$