Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

$3 \frac{1}{2 + 0}$

Step 3

Simplify the answer.

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Step 3.1

Add $2$ and $0$ .

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

Step 3.2

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$