Calculus Examples

$lim_{x \to 8} \frac{1 + 3 e^{2 x} + 2 e^{3 x}}{4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 1

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

$\frac{lim_{x \to 8} 1 + 3 e^{2 x} + 2 e^{3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 2

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

$\frac{lim_{x \to 8} 1 + lim_{x \to 8} 3 e^{2 x} + lim_{x \to 8} 2 e^{3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 3

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

$\frac{1 + lim_{x \to 8} 3 e^{2 x} + lim_{x \to 8} 2 e^{3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 + 3 lim_{x \to 8} e^{2 x} + lim_{x \to 8} 2 e^{3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 5

Move the limit into the exponent.

$\frac{1 + 3 e^{lim_{x \to 8} 2 x} + lim_{x \to 8} 2 e^{3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 6

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

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + lim_{x \to 8} 2 e^{3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 7

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

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 lim_{x \to 8} e^{3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 8

Move the limit into the exponent.

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{lim_{x \to 8} 3 x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 9

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{lim_{x \to 8} 4 + 5 e^{2 x} + 7 e^{3 x}}$

Step 10

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

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{lim_{x \to 8} 4 + lim_{x \to 8} 5 e^{2 x} + lim_{x \to 8} 7 e^{3 x}}$

Step 11

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{4 + lim_{x \to 8} 5 e^{2 x} + lim_{x \to 8} 7 e^{3 x}}$

Step 12

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{4 + 5 lim_{x \to 8} e^{2 x} + lim_{x \to 8} 7 e^{3 x}}$

Step 13

Move the limit into the exponent.

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{4 + 5 e^{lim_{x \to 8} 2 x} + lim_{x \to 8} 7 e^{3 x}}$

Step 14

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

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{4 + 5 e^{2 lim_{x \to 8} x} + lim_{x \to 8} 7 e^{3 x}}$

Step 15

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{4 + 5 e^{2 lim_{x \to 8} x} + 7 lim_{x \to 8} e^{3 x}}$

Step 16

Move the limit into the exponent.

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{4 + 5 e^{2 lim_{x \to 8} x} + 7 e^{lim_{x \to 8} 3 x}}$

Step 17

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 + 3 e^{2 lim_{x \to 8} x} + 2 e^{3 lim_{x \to 8} x}}{4 + 5 e^{2 lim_{x \to 8} x} + 7 e^{3 lim_{x \to 8} x}}$

Step 18

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

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Step 18.1

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

$\frac{1 + 3 e^{2 \cdot 8} + 2 e^{3 lim_{x \to 8} x}}{4 + 5 e^{2 lim_{x \to 8} x} + 7 e^{3 lim_{x \to 8} x}}$

Step 18.2

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

$\frac{1 + 3 e^{2 \cdot 8} + 2 e^{3 \cdot 8}}{4 + 5 e^{2 lim_{x \to 8} x} + 7 e^{3 lim_{x \to 8} x}}$

Step 18.3

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

$\frac{1 + 3 e^{2 \cdot 8} + 2 e^{3 \cdot 8}}{4 + 5 e^{2 \cdot 8} + 7 e^{3 lim_{x \to 8} x}}$

Step 18.4

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

$\frac{1 + 3 e^{2 \cdot 8} + 2 e^{3 \cdot 8}}{4 + 5 e^{2 \cdot 8} + 7 e^{3 \cdot 8}}$

Step 19

Simplify the answer.

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Step 19.1

Simplify the numerator.

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Step 19.1.1

Multiply $2$ by $8$ .

$\frac{1 + 3 e^{16} + 2 e^{3 \cdot 8}}{4 + 5 e^{2 \cdot 8} + 7 e^{3 \cdot 8}}$

Step 19.1.2

Multiply $3$ by $8$ .

$\frac{1 + 3 e^{16} + 2 e^{24}}{4 + 5 e^{2 \cdot 8} + 7 e^{3 \cdot 8}}$

Step 19.2

Simplify the denominator.

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Step 19.2.1

Multiply $2$ by $8$ .

$\frac{1 + 3 e^{16} + 2 e^{24}}{4 + 5 e^{16} + 7 e^{3 \cdot 8}}$

Step 19.2.2

Multiply $3$ by $8$ .

$\frac{1 + 3 e^{16} + 2 e^{24}}{4 + 5 e^{16} + 7 e^{24}}$

Step 20

The result can be shown in multiple forms.

Exact Form:

$\frac{1 + 3 e^{16} + 2 e^{24}}{4 + 5 e^{16} + 7 e^{24}}$

Decimal Form:

$0.28578957 \dots$