Precalculus Examples

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

Step 1

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

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

Step 2

Since the function $e^{6 x}$ approaches $\infty$ , the positive constant $3$ times the function also approaches $\infty$ .

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Step 2.1

Consider the limit with the constant multiple $3$ removed.

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

Step 2.2

Since the exponent $6 x$ approaches $\infty$ , the quantity $e^{6 x}$ approaches $\infty$ .

$\infty - lim_{x \to \infty} 2$

Step 3

Evaluate the limit.

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Step 3.1

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

$\infty - 1 \cdot 2$

Step 3.2

Simplify the answer.

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Step 3.2.1

Multiply $- 1$ by $2$ .

$\infty - 2$

Step 3.2.2

Infinity plus or minus a number is infinity.

$\infty$