Calculus Examples

$lim_{x \to 3} \frac{x^{2 - 9}}{x^{2 - 5 x + 6}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Cancel the common factor of $x^{2 - 9}$ and $x^{2 - 5 x + 6}$ .

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Step 1.1.1

Factor $x^{2 - 5 x + 6}$ out of $x^{2 - 9}$ .

$lim_{x \to 3} \frac{x^{2 - 5 x + 6} x^{- 7 - (2 - 5 x + 6)}}{x^{2 - 5 x + 6}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Multiply by $1$ .

$lim_{x \to 3} \frac{x^{2 - 5 x + 6} x^{- 7 - (2 - 5 x + 6)}}{x^{2 - 5 x + 6} \cdot 1}$

Step 1.1.2.2

Cancel the common factor.

$lim_{x \to 3} \frac{x^{2 - 5 x + 6} x^{- 7 - (2 - 5 x + 6)}}{x^{2 - 5 x + 6} \cdot 1}$

Step 1.1.2.3

Rewrite the expression.

$lim_{x \to 3} \frac{x^{- 7 - (2 - 5 x + 6)}}{1}$

Step 1.1.2.4

Divide $x^{- 7 - (2 - 5 x + 6)}$ by $1$ .

$lim_{x \to 3} x^{- 7 - (2 - 5 x + 6)}$

Step 1.2

Add $2$ and $6$ .

$lim_{x \to 3} x^{- 7 - (- 5 x + 8)}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $x^{- 7 - (- 5 x + 8)}$ as $e^{\ln (x^{- 7 - (- 5 x + 8)})}$ .

$lim_{x \to 3} e^{\ln (x^{- 7 - (- 5 x + 8)})}$

Step 2.2

Expand $\ln (x^{- 7 - (- 5 x + 8)})$ by moving $- 7 - (- 5 x + 8)$ outside the logarithm.

$lim_{x \to 3} e^{(- 7 - (- 5 x + 8)) \ln (x)}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$e^{lim_{x \to 3} (- 7 - (- 5 x + 8)) \ln (x)}$

Step 3.2

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

$e^{lim_{x \to 3} - 7 - (- 5 x + 8) \cdot lim_{x \to 3} \ln (x)}$

Step 3.3

Move the limit inside the logarithm.

$e^{lim_{x \to 3} - 7 - (- 5 x + 8) \cdot \ln (lim_{x \to 3} x)}$

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 $- 7 - (- 5 x + 8)$ by plugging in $3$ for $x$ .

$e^{(- 7 - (- 5 \cdot 3 + 8)) \cdot \ln (lim_{x \to 3} x)}$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $- 5$ by $3$ .

$e^{(- 7 - (- 15 + 8)) \cdot \ln (lim_{x \to 3} x)}$

Step 4.2.2

Add $- 15$ and $8$ .

$e^{(- 7 - - 7) \cdot \ln (lim_{x \to 3} x)}$

Step 4.2.3

Multiply $- 1$ by $- 7$ .

$e^{(- 7 + 7) \cdot \ln (lim_{x \to 3} x)}$

Step 4.3

Add $- 7$ and $7$ .

$e^{0 \cdot \ln (lim_{x \to 3} x)}$

Step 4.4

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

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

Step 5

Simplify the answer.

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Step 5.1

Simplify $0 \cdot \ln (3)$ by moving $0$ inside the logarithm.

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

Step 5.2

Exponentiation and log are inverse functions.

$3^{0}$

Step 5.3

Anything raised to $0$ is $1$ .

$1$