Calculus Examples

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

Step 1

Simplify the limit argument.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Combine factors.

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Step 1.2.1

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

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

Step 1.2.2

Combine $\frac{3}{x^{2}}$ and $e^{x}$ .

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 2.2

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.

$lim_{x \to \infty} \frac{3 e^{x}}{x^{2}} = lim_{x \to \infty} \frac{\frac{d}{d x} [3 e^{x}]}{\frac{d}{d x} [x^{2}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x}$ with respect to $x$ is $3 \frac{d}{d x} [e^{x}]$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.2

$lim_{x \to \infty} \frac{3 e^{x}}{2 x} = lim_{x \to \infty} \frac{\frac{d}{d x} [3 e^{x}]}{\frac{d}{d x} [2 x]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x}$ with respect to $x$ is $3 \frac{d}{d x} [e^{x}]$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Multiply $2$ by $1$ .

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

Step 4

Since the function $e^{x}$ approaches $\infty$ , the positive constant $\frac{3}{2}$ times the function also approaches $\infty$ .

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Step 4.1

Consider the limit with the constant multiple $\frac{3}{2}$ removed.

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

Step 4.2

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

$\infty$