Calculus Examples

lim_{x \to - \infty} \frac{x}{2 x - 3}

$lim_{x \to - \infty} \frac{x}{2 x - 3}$

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} x}{lim_{x \to - \infty} 2 x - 3}

$\frac{lim_{x \to - \infty} x}{lim_{x \to - \infty} 2 x - 3}$

Step 1.2

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

\frac{- \infty}{lim_{x \to - \infty} 2 x - 3}

$\frac{- \infty}{lim_{x \to - \infty} 2 x - 3}$

Step 1.3

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

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

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

Step 1.4

Infinity divided by infinity is undefined.

Undefined

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

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

Step 2

Since

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

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

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

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} [x]}{\frac{d}{d x} [2 x - 3]}

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

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

lim_{x \to - \infty} \frac{1}{\frac{d}{d x} [2 x - 3]}

$lim_{x \to - \infty} \frac{1}{\frac{d}{d x} [2 x - 3]}$

Step 3.3

By the Sum Rule, the derivative of

2 x - 3

$2 x - 3$ with respect to

x

$x$ is

\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$ .

lim_{x \to - \infty} \frac{1}{\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]}

$lim_{x \to - \infty} \frac{1}{\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]}$

Step 3.4

Evaluate

\frac{d}{d x} [2 x]

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

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Step 3.4.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x

$2 x$ with respect to

x

$x$ is

2 \frac{d}{d x} [x]

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

lim_{x \to - \infty} \frac{1}{2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]}

$lim_{x \to - \infty} \frac{1}{2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]}$

Step 3.4.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

lim_{x \to - \infty} \frac{1}{2 \cdot 1 + \frac{d}{d x} [- 3]}

$lim_{x \to - \infty} \frac{1}{2 \cdot 1 + \frac{d}{d x} [- 3]}$

Step 3.4.3

Multiply

2

$2$ by

1

$1$ .

lim_{x \to - \infty} \frac{1}{2 + \frac{d}{d x} [- 3]}

$lim_{x \to - \infty} \frac{1}{2 + \frac{d}{d x} [- 3]}$

lim_{x \to - \infty} \frac{1}{2 + \frac{d}{d x} [- 3]}

$lim_{x \to - \infty} \frac{1}{2 + \frac{d}{d x} [- 3]}$

Step 3.5

Since

- 3

$- 3$ is constant with respect to

x

$x$ , the derivative of

- 3

$- 3$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 3.6

Add

2

$2$ and

0

$0$ .

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

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

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

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

Step 4

Evaluate the limit of

\frac{1}{2}

$\frac{1}{2}$ which is constant as

x

$x$ approaches

- \infty

$- \infty$ .

\frac{1}{2}

$\frac{1}{2}$