Calculus Examples

$lim_{x \to \infty} \frac{e^{x^{2}}}{x^{4}}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \infty} e^{x^{2}}}{lim_{x \to \infty} x^{4}}$

Step 1.2

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

$\frac{\infty}{lim_{x \to \infty} x^{4}}$

Step 1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{\infty}{\infty}$

Step 1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{\infty}$

Step 2

Since $\frac{\infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \infty} \frac{e^{x^{2}}}{x^{4}} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x^{2}}]}{\frac{d}{d x} [x^{4}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x^{2}}]}{\frac{d}{d x} [x^{4}]}$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$lim_{x \to \infty} \frac{\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [x^{4}]}$

Step 3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to \infty} \frac{e^{u} \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [x^{4}]}$

Step 3.2.3

Replace all occurrences of $u$ with $x^{2}$ .

$lim_{x \to \infty} \frac{e^{x^{2}} \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [x^{4}]}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to \infty} \frac{e^{x^{2}} (2 x)}{\frac{d}{d x} [x^{4}]}$

Step 3.4

Simplify.

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Step 3.4.1

Reorder the factors of $e^{x^{2}} (2 x)$ .

$lim_{x \to \infty} \frac{2 e^{x^{2}} x}{\frac{d}{d x} [x^{4}]}$

Step 3.4.2

Reorder factors in $2 e^{x^{2}} x$ .

$lim_{x \to \infty} \frac{2 x e^{x^{2}}}{\frac{d}{d x} [x^{4}]}$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$lim_{x \to \infty} \frac{2 x e^{x^{2}}}{4 x^{3}}$

Step 4

Reduce.

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Step 4.1

Cancel the common factor of $2$ and $4$ .

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Step 4.1.1

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

$lim_{x \to \infty} \frac{2 (x e^{x^{2}})}{4 x^{3}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $4 x^{3}$ .

$lim_{x \to \infty} \frac{2 (x e^{x^{2}})}{2 (2 x^{3})}$

Step 4.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{2 (x e^{x^{2}})}{2 (2 x^{3})}$

Step 4.1.2.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 4.2.1

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

$lim_{x \to \infty} \frac{x (e^{x^{2}})}{2 x^{3}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x$ out of $2 x^{3}$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 5

Apply L'Hospital's rule.

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Step 5.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 5.1.2

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

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

Step 5.1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{\infty}{\infty}$

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{\infty}$

Step 5.2

$lim_{x \to \infty} \frac{e^{x^{2}}}{2 x^{2}} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x^{2}}]}{\frac{d}{d x} [2 x^{2}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

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

Step 5.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 5.3.2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$lim_{x \to \infty} \frac{\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [2 x^{2}]}$

Step 5.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to \infty} \frac{e^{u} \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [2 x^{2}]}$

Step 5.3.2.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 5.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to \infty} \frac{e^{x^{2}} (2 x)}{\frac{d}{d x} [2 x^{2}]}$

Step 5.3.4

Simplify.

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Step 5.3.4.1

Reorder the factors of $e^{x^{2}} (2 x)$ .

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

Step 5.3.4.2

Reorder factors in $2 e^{x^{2}} x$ .

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

Step 5.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

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

Step 5.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 5.3.7

Multiply $2$ by $2$ .

$lim_{x \to \infty} \frac{2 x e^{x^{2}}}{4 x}$

Step 5.4

Reduce.

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Step 5.4.1

Cancel the common factor of $2$ and $4$ .

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Step 5.4.1.1

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

$lim_{x \to \infty} \frac{2 (x e^{x^{2}})}{4 x}$

Step 5.4.1.2

Cancel the common factors.

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Step 5.4.1.2.1

Factor $2$ out of $4 x$ .

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

Step 5.4.1.2.2

Cancel the common factor.

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

Step 5.4.1.2.3

Rewrite the expression.

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

Step 5.4.2

Cancel the common factor of $x$ .

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Step 5.4.2.1

Cancel the common factor.

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

Step 5.4.2.2

Rewrite the expression.

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

Step 6

Since the function $e^{x^{2}}$ approaches $\infty$ , the positive constant $\frac{1}{2}$ times the function also approaches $\infty$ .

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Step 6.1

Consider the limit with the constant multiple $\frac{1}{2}$ removed.

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

Step 6.2

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

$\infty$