Calculus Examples

$lim_{x \to - \infty} \frac{3 + 2^{x}}{4 - 5^{x}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to - \infty} 3 + 2^{x}}{lim_{x \to - \infty} 4 - 5^{x}}$

Step 1.2

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

$\frac{lim_{x \to - \infty} 3 + lim_{x \to - \infty} 2^{x}}{lim_{x \to - \infty} 4 - 5^{x}}$

Step 1.3

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

$\frac{3 + lim_{x \to - \infty} 2^{x}}{lim_{x \to - \infty} 4 - 5^{x}}$

Step 2

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

$\frac{3 + 0}{lim_{x \to - \infty} 4 - 5^{x}}$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{3 + 0}{lim_{x \to - \infty} 4 - lim_{x \to - \infty} 5^{x}}$

Step 3.2

Evaluate the limit of $4$ which is constant as $x$ approaches $- \infty$ .

$\frac{3 + 0}{4 - lim_{x \to - \infty} 5^{x}}$

Step 4

Since the exponent $x$ approaches $- \infty$ , the quantity $5^{x}$ approaches $0$ .

$\frac{3 + 0}{4 - 0}$

Step 5

Simplify the answer.

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Step 5.1

Add $3$ and $0$ .

$\frac{3}{4 - 0}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $0$ .

$\frac{3}{4 + 0}$

Step 5.2.2

Add $4$ and $0$ .

$\frac{3}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$