Calculus Examples

$lim_{x \to \infty} \frac{6}{3 + e^{- 2 x}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 lim_{x \to \infty} \frac{1}{3 + e^{- 2 x}}$

Step 1.2

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

$6 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} 3 + e^{- 2 x}}$

Step 1.3

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

$6 \frac{1}{lim_{x \to \infty} 3 + e^{- 2 x}}$

Step 1.4

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

$6 \frac{1}{lim_{x \to \infty} 3 + lim_{x \to \infty} e^{- 2 x}}$

Step 1.5

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

$6 \frac{1}{3 + lim_{x \to \infty} e^{- 2 x}}$

Step 2

Since the exponent $- 2 x$ approaches $- \infty$ , the quantity $e^{- 2 x}$ approaches $0$ .

$6 \frac{1}{3 + 0}$

Step 3

Simplify the answer.

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Step 3.1

Add $3$ and $0$ .

$6 (\frac{1}{3})$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Factor $3$ out of $6$ .

$3 (2) \frac{1}{3}$

Step 3.2.2

Cancel the common factor.

$3 \cdot 2 \frac{1}{3}$

Step 3.2.3

Rewrite the expression.

$2$