Calculus Examples

$lim_{x \to 8} {(\frac{2 x - 3}{2 x + 5})}^{2 x + 1}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(\frac{2 x - 3}{2 x + 5})}^{2 x + 1}$ as $e^{\ln ({(\frac{2 x - 3}{2 x + 5})}^{2 x + 1})}$ .

$lim_{x \to 8} e^{\ln ({(\frac{2 x - 3}{2 x + 5})}^{2 x + 1})}$

Step 1.2

Expand $\ln ({(\frac{2 x - 3}{2 x + 5})}^{2 x + 1})$ by moving $2 x + 1$ outside the logarithm.

$lim_{x \to 8} e^{(2 x + 1) \ln (\frac{2 x - 3}{2 x + 5})}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 8} (2 x + 1) \ln (\frac{2 x - 3}{2 x + 5})}$

Step 2.2

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

$e^{lim_{x \to 8} 2 x + 1 \cdot lim_{x \to 8} \ln (\frac{2 x - 3}{2 x + 5})}$

Step 2.3

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

$e^{(lim_{x \to 8} 2 x + lim_{x \to 8} 1) \cdot lim_{x \to 8} \ln (\frac{2 x - 3}{2 x + 5})}$

Step 2.4

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

$e^{(2 lim_{x \to 8} x + lim_{x \to 8} 1) \cdot lim_{x \to 8} \ln (\frac{2 x - 3}{2 x + 5})}$

Step 2.5

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

$e^{(2 lim_{x \to 8} x + 1) \cdot lim_{x \to 8} \ln (\frac{2 x - 3}{2 x + 5})}$

Step 2.6

Move the limit inside the logarithm.

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (lim_{x \to 8} \frac{2 x - 3}{2 x + 5})}$

Step 2.7

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

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (\frac{lim_{x \to 8} 2 x - 3}{lim_{x \to 8} 2 x + 5})}$

Step 2.8

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

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (\frac{lim_{x \to 8} 2 x - lim_{x \to 8} 3}{lim_{x \to 8} 2 x + 5})}$

Step 2.9

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

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (\frac{2 lim_{x \to 8} x - lim_{x \to 8} 3}{lim_{x \to 8} 2 x + 5})}$

Step 2.10

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

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 3}{lim_{x \to 8} 2 x + 5})}$

Step 2.11

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

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 3}{lim_{x \to 8} 2 x + lim_{x \to 8} 5})}$

Step 2.12

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

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 3}{2 lim_{x \to 8} x + lim_{x \to 8} 5})}$

Step 2.13

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$e^{(2 lim_{x \to 8} x + 1) \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 3}{2 lim_{x \to 8} x + 5})}$

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 $x$ .

$e^{(2 \cdot 8 + 1) \cdot \ln (\frac{2 lim_{x \to 8} x - 1 \cdot 3}{2 lim_{x \to 8} x + 5})}$

Step 3.2

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

$e^{(2 \cdot 8 + 1) \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 3}{2 lim_{x \to 8} x + 5})}$

Step 3.3

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

$e^{(2 \cdot 8 + 1) \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 3}{2 \cdot 8 + 5})}$

Step 4

Simplify the answer.

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Step 4.1

Multiply $2$ by $8$ .

$e^{(16 + 1) \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 3}{2 \cdot 8 + 5})}$

Step 4.2

Add $16$ and $1$ .

$e^{17 \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 3}{2 \cdot 8 + 5})}$

Step 4.3

Simplify $17 \cdot \ln (\frac{2 \cdot 8 - 1 \cdot 3}{2 \cdot 8 + 5})$ by moving $17$ inside the logarithm.

$e^{\ln ({(\frac{2 \cdot 8 - 1 \cdot 3}{2 \cdot 8 + 5})}^{17})}$

Step 4.4

Exponentiation and log are inverse functions.

${(\frac{2 \cdot 8 - 1 \cdot 3}{2 \cdot 8 + 5})}^{17}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $2$ by $8$ .

${(\frac{16 - 1 \cdot 3}{2 \cdot 8 + 5})}^{17}$

Step 4.5.2

Multiply $- 1$ by $3$ .

${(\frac{16 - 3}{2 \cdot 8 + 5})}^{17}$

Step 4.5.3

Subtract $3$ from $16$ .

${(\frac{13}{2 \cdot 8 + 5})}^{17}$

Step 4.6

Simplify the denominator.

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Step 4.6.1

Multiply $2$ by $8$ .

${(\frac{13}{16 + 5})}^{17}$

Step 4.6.2

Add $16$ and $5$ .

${(\frac{13}{21})}^{17}$

Step 4.7

Apply the product rule to $\frac{13}{21}$ .

$\frac{13^{17}}{21^{17}}$

Step 4.8

Raise $13$ to the power of $17$ .

$\frac{8650415919381340000}{21^{17}}$

Step 4.9

Raise $21$ to the power of $17$ .

$\frac{8650415919381340000}{30041942495081700000000}$

Step 4.10

Cancel the common factor of $8650415919381340000$ and $30041942495081700000000$ .

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Step 4.10.1

Factor $20000$ out of $8650415919381340000$ .

$\frac{20000 (432520795969067)}{30041942495081700000000}$

Step 4.10.2

Cancel the common factors.

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Step 4.10.2.1

Factor $20000$ out of $30041942495081700000000$ .

$\frac{20000 \cdot 432520795969067}{20000 \cdot 1502097124754085000}$

Step 4.10.2.2

Cancel the common factor.

$\frac{20000 \cdot 432520795969067}{20000 \cdot 1502097124754085000}$

Step 4.10.2.3

Rewrite the expression.

$\frac{432520795969067}{1502097124754085000}$