Calculus Examples

$lim_{x \to \infty} \frac{11 x^{2} - 2 x + 8}{2 - x}$

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.

$\frac{lim_{x \to \infty} 11 x^{2} - 2 x + 8}{lim_{x \to \infty} 2 - x}$

Step 1.2

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

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Reorder $2$ and $- x$ .

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

Step 1.3.2

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

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

Step 1.3.3

Infinity divided by infinity is undefined.

Undefined

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

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.

$lim_{x \to \infty} \frac{11 x^{2} - 2 x + 8}{2 - x} = lim_{x \to \infty} \frac{\frac{d}{d x} [11 x^{2} - 2 x + 8]}{\frac{d}{d x} [2 - x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [11 x^{2} - 2 x + 8]}{\frac{d}{d x} [2 - x]}$

Step 3.2

By the Sum Rule, the derivative of $11 x^{2} - 2 x + 8$ with respect to $x$ is $\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [8]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [11 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [8]}{\frac{d}{d x} [2 - x]}$

Step 3.3

Evaluate $\frac{d}{d x} [11 x^{2}]$ .

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Step 3.3.1

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{2}$ with respect to $x$ is $11 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to \infty} \frac{11 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [8]}{\frac{d}{d x} [2 - x]}$

Step 3.3.2

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

$lim_{x \to \infty} \frac{11 (2 x) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [8]}{\frac{d}{d x} [2 - x]}$

Step 3.3.3

Multiply $2$ by $11$ .

$lim_{x \to \infty} \frac{22 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [8]}{\frac{d}{d x} [2 - x]}$

Step 3.4

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 3.4.1

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

$lim_{x \to \infty} \frac{22 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [8]}{\frac{d}{d x} [2 - x]}$

Step 3.4.2

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

$lim_{x \to \infty} \frac{22 x - 2 \cdot 1 + \frac{d}{d x} [8]}{\frac{d}{d x} [2 - x]}$

Step 3.4.3

Multiply $- 2$ by $1$ .

$lim_{x \to \infty} \frac{22 x - 2 + \frac{d}{d x} [8]}{\frac{d}{d x} [2 - x]}$

Step 3.5

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{22 x - 2 + 0}{\frac{d}{d x} [2 - x]}$

Step 3.6

Add $22 x - 2$ and $0$ .

$lim_{x \to \infty} \frac{22 x - 2}{\frac{d}{d x} [2 - x]}$

Step 3.7

By the Sum Rule, the derivative of $2 - x$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- x]$ .

$lim_{x \to \infty} \frac{22 x - 2}{\frac{d}{d x} [2] + \frac{d}{d x} [- x]}$

Step 3.8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{22 x - 2}{0 + \frac{d}{d x} [- x]}$

Step 3.9

Evaluate $\frac{d}{d x} [- x]$ .

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Step 3.9.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{22 x - 2}{0 - \frac{d}{d x} [x]}$

Step 3.9.2

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

$lim_{x \to \infty} \frac{22 x - 2}{0 - 1 \cdot 1}$

Step 3.9.3

Multiply $- 1$ by $1$ .

$lim_{x \to \infty} \frac{22 x - 2}{0 - 1}$

Step 3.10

Subtract $1$ from $0$ .

$lim_{x \to \infty} \frac{22 x - 2}{- 1}$

Step 4

Simplify by multiplying through.

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Step 4.1

Move the negative one from the denominator of $\frac{22 x - 2}{- 1}$ .

$lim_{x \to \infty} - 1 \cdot (22 x - 2)$

Step 4.2

Apply the distributive property.

$lim_{x \to \infty} - 1 \cdot (22 x) - 1 \cdot - 2$

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $22$ .

$lim_{x \to \infty} - 22 x - 1 \cdot - 2$

Step 4.3.2

Multiply $- 1$ by $- 2$ .

$lim_{x \to \infty} - 22 x + 2$

Step 5

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

$- \infty$