Precalculus Examples

$lim_{x \to 8} {(\frac{2 x - 1}{x + 2})}^{\frac{1}{x - 3}}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(\frac{2 x - 1}{x + 2})}^{\frac{1}{x - 3}}$ as $e^{\ln ({(\frac{2 x - 1}{x + 2})}^{\frac{1}{x - 3}})}$ .

$lim_{x \to 8} e^{\ln ({(\frac{2 x - 1}{x + 2})}^{\frac{1}{x - 3}})}$

Step 1.2

Expand $\ln ({(\frac{2 x - 1}{x + 2})}^{\frac{1}{x - 3}})$ by moving $\frac{1}{x - 3}$ outside the logarithm.

$lim_{x \to 8} e^{\frac{1}{x - 3} \ln (\frac{2 x - 1}{x + 2})}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 8} \frac{1}{x - 3} \ln (\frac{2 x - 1}{x + 2})}$

Step 2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$e^{lim_{x \to 8} \frac{1}{x - 3} \cdot lim_{x \to 8} \ln (\frac{2 x - 1}{x + 2})}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$e^{\frac{lim_{x \to 8} 1}{lim_{x \to 8} x - 3} \cdot lim_{x \to 8} \ln (\frac{2 x - 1}{x + 2})}$

Step 2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - 3} \cdot lim_{x \to 8} \ln (\frac{2 x - 1}{x + 2})}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - lim_{x \to 8} 3} \cdot lim_{x \to 8} \ln (\frac{2 x - 1}{x + 2})}$

Step 2.6

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot lim_{x \to 8} \ln (\frac{2 x - 1}{x + 2})}$

Step 2.7

Move the limit inside the logarithm.

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot \ln (lim_{x \to 8} \frac{2 x - 1}{x + 2})}$

Step 2.8

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot \ln (\frac{lim_{x \to 8} 2 x - 1}{lim_{x \to 8} x + 2})}$

Step 2.9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot \ln (\frac{lim_{x \to 8} 2 x - lim_{x \to 8} 1}{lim_{x \to 8} x + 2})}$

Step 2.10

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

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot \ln (\frac{2 lim_{x \to 8} x - lim_{x \to 8} 1}{lim_{x \to 8} x + 2})}$

Step 2.11

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 1}{lim_{x \to 8} x + 2})}$

Step 2.12

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 1}{lim_{x \to 8} x + lim_{x \to 8} 2})}$

Step 2.13

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$e^{\frac{1}{lim_{x \to 8} x - 1 \cdot 3} \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 1}{lim_{x \to 8} x + 2})}$

Step 3

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$e^{\frac{1}{8 - 1 \cdot 3} \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 1}{lim_{x \to 8} x + 2})}$

Step 3.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$e^{\frac{1}{8 - 1 \cdot 3} \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 1}{lim_{x \to 8} x + 2})}$

Step 3.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$e^{\frac{1}{8 - 1 \cdot 3} \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 1}{8 + 2})}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $- 1$ by $3$ .

$e^{\frac{1}{8 - 3} \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 1}{8 + 2})}$

Step 4.1.2

Subtract $3$ from $8$ .

$e^{\frac{1}{5} \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 1}{8 + 2})}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Multiply $2$ by $8$ .

$e^{\frac{1}{5} \cdot \ln (\frac{16 - 1 \cdot 1}{8 + 2})}$

Step 4.2.2

Multiply $- 1$ by $1$ .

$e^{\frac{1}{5} \cdot \ln (\frac{16 - 1}{8 + 2})}$

Step 4.2.3

Subtract $1$ from $16$ .

$e^{\frac{1}{5} \cdot \ln (\frac{15}{8 + 2})}$

Step 4.3

Add $8$ and $2$ .

$e^{\frac{1}{5} \cdot \ln (\frac{15}{10})}$

Step 4.4

Cancel the common factor of $15$ and $10$ .

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Step 4.4.1

Factor $5$ out of $15$ .

$e^{\frac{1}{5} \cdot \ln (\frac{5 (3)}{10})}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $5$ out of $10$ .

$e^{\frac{1}{5} \cdot \ln (\frac{5 \cdot 3}{5 \cdot 2})}$

Step 4.4.2.2

Cancel the common factor.

$e^{\frac{1}{5} \cdot \ln (\frac{5 \cdot 3}{5 \cdot 2})}$

Step 4.4.2.3

Rewrite the expression.

$e^{\frac{1}{5} \cdot \ln (\frac{3}{2})}$

Step 4.5

Combine $\frac{1}{5}$ and $\ln (\frac{3}{2})$ .

$e^{\frac{\ln (\frac{3}{2})}{5}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{\ln (\frac{3}{2})}{5}}$

Decimal Form:

$1.08447177 \dots$