Calculus Examples

$lim_{x \to 8} \frac{15 e^{x} - 18}{3 e^{x}}$

Step 1

Cancel the common factor of $15 e^{x} - 18$ and $3$ .

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Step 1.1

Factor $3$ out of $15 e^{x}$ .

$lim_{x \to 8} \frac{3 (5 e^{x}) - 18}{3 e^{x}}$

Step 1.2

Factor $3$ out of $- 18$ .

$lim_{x \to 8} \frac{3 (5 e^{x}) + 3 \cdot - 6}{3 e^{x}}$

Step 1.3

Factor $3$ out of $3 (5 e^{x}) + 3 (- 6)$ .

$lim_{x \to 8} \frac{3 (5 e^{x} - 6)}{3 e^{x}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $3$ out of $3 e^{x}$ .

$lim_{x \to 8} \frac{3 (5 e^{x} - 6)}{3 (e^{x})}$

Step 1.4.2

Cancel the common factor.

$lim_{x \to 8} \frac{3 (5 e^{x} - 6)}{3 e^{x}}$

Step 1.4.3

Rewrite the expression.

$lim_{x \to 8} \frac{5 e^{x} - 6}{e^{x}}$

Step 2

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

$\frac{lim_{x \to 8} 5 e^{x} - 6}{lim_{x \to 8} e^{x}}$

Step 3

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

$\frac{lim_{x \to 8} 5 e^{x} - lim_{x \to 8} 6}{lim_{x \to 8} e^{x}}$

Step 4

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

$\frac{5 lim_{x \to 8} e^{x} - lim_{x \to 8} 6}{lim_{x \to 8} e^{x}}$

Step 5

Move the limit into the exponent.

$\frac{5 e^{lim_{x \to 8} x} - lim_{x \to 8} 6}{lim_{x \to 8} e^{x}}$

Step 6

Evaluate the limit of $6$ which is constant as $x$ approaches $8$ .

$\frac{5 e^{lim_{x \to 8} x} - 1 \cdot 6}{lim_{x \to 8} e^{x}}$

Step 7

Move the limit into the exponent.

$\frac{5 e^{lim_{x \to 8} x} - 1 \cdot 6}{e^{lim_{x \to 8} x}}$

Step 8

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

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Step 8.1

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

$\frac{5 e^{8} - 1 \cdot 6}{e^{lim_{x \to 8} x}}$

Step 8.2

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

$\frac{5 e^{8} - 1 \cdot 6}{e^{8}}$

Step 9

Multiply $- 1$ by $6$ .

$\frac{5 e^{8} - 6}{e^{8}}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{5 e^{8} - 6}{e^{8}}$

Decimal Form:

$4.99798722 \dots$