Calculus Examples

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

Step 1

Cancel the common factor of $e^{x^{2}}$ and $e^{2 x}$ .

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Step 1.1

Factor $e^{2 x}$ out of $e^{x^{2}}$ .

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

Multiply by $1$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.2.4

Divide $e^{x^{2} - 2 x}$ by $1$ .

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

Step 2

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

$\infty$