Calculus Examples

$lim_{x \to 3} \frac{x^{2} - 2 x - 3}{x^{2 - 6 x + 9}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 3} x^{2} - 2 x - 3}{lim_{x \to 3} x^{2 - 6 x + 9}}$

Step 1.2

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

$\frac{lim_{x \to 3} x^{2} - lim_{x \to 3} 2 x - lim_{x \to 3} 3}{lim_{x \to 3} x^{2 - 6 x + 9}}$

Step 1.3

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

$\frac{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 2 x - lim_{x \to 3} 3}{lim_{x \to 3} x^{2 - 6 x + 9}}$

Step 1.4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - lim_{x \to 3} 3}{lim_{x \to 3} x^{2 - 6 x + 9}}$

Step 1.5

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

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} x^{2 - 6 x + 9}}$

Step 1.6

Add $2$ and $9$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} x^{- 6 x + 11}}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $x^{- 6 x + 11}$ as $e^{\ln (x^{- 6 x + 11})}$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} e^{\ln (x^{- 6 x + 11})}}$

Step 2.2

Expand $\ln (x^{- 6 x + 11})$ by moving $- 6 x + 11$ outside the logarithm.

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} e^{(- 6 x + 11) \ln (x)}}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{e^{lim_{x \to 3} (- 6 x + 11) \ln (x)}}$

Step 3.2

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

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{e^{lim_{x \to 3} - 6 x + 11 \cdot lim_{x \to 3} \ln (x)}}$

Step 3.3

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

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{e^{(- lim_{x \to 3} 6 x + lim_{x \to 3} 11) \cdot lim_{x \to 3} \ln (x)}}$

Step 3.4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{e^{(- 6 lim_{x \to 3} x + lim_{x \to 3} 11) \cdot lim_{x \to 3} \ln (x)}}$

Step 3.5

Evaluate the limit of $11$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{e^{(- 6 lim_{x \to 3} x + 11) \cdot lim_{x \to 3} \ln (x)}}$

Step 3.6

Move the limit inside the logarithm.

$\frac{{(lim_{x \to 3} x)}^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{e^{(- 6 lim_{x \to 3} x + 11) \cdot \ln (lim_{x \to 3} x)}}$

Step 4

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

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Step 4.1

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

$\frac{3^{2} - 2 lim_{x \to 3} x - 1 \cdot 3}{e^{(- 6 lim_{x \to 3} x + 11) \cdot \ln (lim_{x \to 3} x)}}$

Step 4.2

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

$\frac{3^{2} - 2 \cdot 3 - 1 \cdot 3}{e^{(- 6 lim_{x \to 3} x + 11) \cdot \ln (lim_{x \to 3} x)}}$

Step 4.3

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

$\frac{3^{2} - 2 \cdot 3 - 1 \cdot 3}{e^{(- 6 \cdot 3 + 11) \cdot \ln (lim_{x \to 3} x)}}$

Step 4.4

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

$\frac{3^{2} - 2 \cdot 3 - 1 \cdot 3}{e^{(- 6 \cdot 3 + 11) \cdot \ln (3)}}$

Step 5

Simplify the answer.

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Step 5.1

Move $e^{(- 6 \cdot 3 + 11) \cdot \ln (3)}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$(3^{2} - 2 \cdot 3 - 1 \cdot 3) e^{- (- 6 \cdot 3 + 11) \cdot \ln (3)}$

Step 5.2

Simplify each term.

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Step 5.2.1

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

$(9 - 2 \cdot 3 - 1 \cdot 3) e^{- (- 6 \cdot 3 + 11) \cdot \ln (3)}$

Step 5.2.2

Multiply $- 2$ by $3$ .

$(9 - 6 - 1 \cdot 3) e^{- (- 6 \cdot 3 + 11) \cdot \ln (3)}$

Step 5.2.3

Multiply $- 1$ by $3$ .

$(9 - 6 - 3) e^{- (- 6 \cdot 3 + 11) \cdot \ln (3)}$

Step 5.3

Subtract $6$ from $9$ .

$(3 - 3) e^{- (- 6 \cdot 3 + 11) \cdot \ln (3)}$

Step 5.4

Subtract $3$ from $3$ .

$0 e^{- (- 6 \cdot 3 + 11) \cdot \ln (3)}$

Step 5.5

Multiply $- 6$ by $3$ .

$0 e^{- (- 18 + 11) \cdot \ln (3)}$

Step 5.6

Add $- 18$ and $11$ .

$0 e^{- - 7 \cdot \ln (3)}$

Step 5.7

Multiply $- 1$ by $- 7$ .

$0 e^{7 \cdot \ln (3)}$

Step 5.8

Simplify $7 \cdot \ln (3)$ by moving $7$ inside the logarithm.

$0 e^{\ln (3^{7})}$

Step 5.9

Exponentiation and log are inverse functions.

$0 \cdot 3^{7}$

Step 5.10

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

$0 \cdot 2187$

Step 5.11

Multiply $0$ by $2187$ .

$0$