Calculus Examples

$lim_{x \to \infty} \frac{6 x}{15 x - 8}$

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.

$\frac{lim_{x \to \infty} 6 x}{lim_{x \to \infty} 15 x - 8}$

Step 1.2

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

$\frac{\infty}{lim_{x \to \infty} 15 x - 8}$

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.

$lim_{x \to \infty} \frac{6 x}{15 x - 8} = lim_{x \to \infty} \frac{\frac{d}{d x} [6 x]}{\frac{d}{d x} [15 x - 8]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [6 x]}{\frac{d}{d x} [15 x - 8]}$

Step 3.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{6 \frac{d}{d x} [x]}{\frac{d}{d x} [15 x - 8]}$

Step 3.3

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

$lim_{x \to \infty} \frac{6 \cdot 1}{\frac{d}{d x} [15 x - 8]}$

Step 3.4

Multiply $6$ by $1$ .

$lim_{x \to \infty} \frac{6}{\frac{d}{d x} [15 x - 8]}$

Step 3.5

By the Sum Rule, the derivative of $15 x - 8$ with respect to $x$ is $\frac{d}{d x} [15 x] + \frac{d}{d x} [- 8]$ .

$lim_{x \to \infty} \frac{6}{\frac{d}{d x} [15 x] + \frac{d}{d x} [- 8]}$

Step 3.6

Evaluate $\frac{d}{d x} [15 x]$ .

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Step 3.6.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x$ with respect to $x$ is $15 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{6}{15 \frac{d}{d x} [x] + \frac{d}{d x} [- 8]}$

Step 3.6.2

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

$lim_{x \to \infty} \frac{6}{15 \cdot 1 + \frac{d}{d x} [- 8]}$

Step 3.6.3

Multiply $15$ by $1$ .

$lim_{x \to \infty} \frac{6}{15 + \frac{d}{d x} [- 8]}$

Step 3.7

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

$lim_{x \to \infty} \frac{6}{15 + 0}$

Step 3.8

Add $15$ and $0$ .

$lim_{x \to \infty} \frac{6}{15}$

Step 4

Cancel the common factor of $6$ and $15$ .

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Step 4.1

Factor $3$ out of $6$ .

$lim_{x \to \infty} \frac{3 (2)}{15}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $15$ .

$lim_{x \to \infty} \frac{3 \cdot 2}{3 \cdot 5}$

Step 4.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{3 \cdot 2}{3 \cdot 5}$

Step 4.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{2}{5}$

Step 5

Evaluate the limit of $\frac{2}{5}$ which is constant as $x$ approaches $\infty$ .

$\frac{2}{5}$