Calculus Examples

$lim_{x \to - \infty} \frac{9 x^{2} + 10 x - 1}{8 x^{2} - 5 x}$

Step 1

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

Tap for more steps...

Step 1.1

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

$\frac{lim_{x \to - \infty} 9 x^{2} + 10 x - 1}{lim_{x \to - \infty} 8 x^{2} - 5 x}$

Step 1.2

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

$\frac{\infty}{lim_{x \to - \infty} 8 x^{2} - 5 x}$

Step 1.3

The limit at negative infinity of a polynomial of even degree 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.

$lim_{x \to - \infty} \frac{9 x^{2} + 10 x - 1}{8 x^{2} - 5 x} = lim_{x \to - \infty} \frac{\frac{d}{d x} [9 x^{2} + 10 x - 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to - \infty} \frac{\frac{d}{d x} [9 x^{2} + 10 x - 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.2

By the Sum Rule, the derivative of $9 x^{2} + 10 x - 1$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to - \infty} \frac{\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.3

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

Tap for more steps...

Step 3.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - \infty} \frac{9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.3.2

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

$lim_{x \to - \infty} \frac{9 (2 x) + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.3.3

Multiply $2$ by $9$ .

$lim_{x \to - \infty} \frac{18 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.4

Evaluate $\frac{d}{d x} [10 x]$ .

Tap for more steps...

Step 3.4.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$lim_{x \to - \infty} \frac{18 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.4.2

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

$lim_{x \to - \infty} \frac{18 x + 10 \cdot 1 + \frac{d}{d x} [- 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.4.3

Multiply $10$ by $1$ .

$lim_{x \to - \infty} \frac{18 x + 10 + \frac{d}{d x} [- 1]}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to - \infty} \frac{18 x + 10 + 0}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.6

Add $18 x + 10$ and $0$ .

$lim_{x \to - \infty} \frac{18 x + 10}{\frac{d}{d x} [8 x^{2} - 5 x]}$

Step 3.7

By the Sum Rule, the derivative of $8 x^{2} - 5 x$ with respect to $x$ is $\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- 5 x]$ .

$lim_{x \to - \infty} \frac{18 x + 10}{\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- 5 x]}$

Step 3.8

Evaluate $\frac{d}{d x} [8 x^{2}]$ .

Tap for more steps...

Step 3.8.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - \infty} \frac{18 x + 10}{8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x]}$

Step 3.8.2

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

$lim_{x \to - \infty} \frac{18 x + 10}{8 (2 x) + \frac{d}{d x} [- 5 x]}$

Step 3.8.3

Multiply $2$ by $8$ .

$lim_{x \to - \infty} \frac{18 x + 10}{16 x + \frac{d}{d x} [- 5 x]}$

Step 3.9

Evaluate $\frac{d}{d x} [- 5 x]$ .

Tap for more steps...

Step 3.9.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$lim_{x \to - \infty} \frac{18 x + 10}{16 x - 5 \frac{d}{d x} [x]}$

Step 3.9.2

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

$lim_{x \to - \infty} \frac{18 x + 10}{16 x - 5 \cdot 1}$

Step 3.9.3

Multiply $- 5$ by $1$ .

$lim_{x \to - \infty} \frac{18 x + 10}{16 x - 5}$

Step 4

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

$lim_{x \to - \infty} \frac{\frac{18 x}{x} + \frac{10}{x}}{\frac{16 x}{x} + \frac{- 5}{x}}$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{18 x}{x} + \frac{10}{x}}{\frac{16 x}{x} + \frac{- 5}{x}}$

Step 5.1.2

Divide $18$ by $1$ .

$lim_{x \to - \infty} \frac{18 + \frac{10}{x}}{\frac{16 x}{x} + \frac{- 5}{x}}$

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{18 + \frac{10}{x}}{\frac{16 x}{x} + \frac{- 5}{x}}$

Step 5.2.1.2

Divide $16$ by $1$ .

$lim_{x \to - \infty} \frac{18 + \frac{10}{x}}{16 + \frac{- 5}{x}}$

Step 5.2.2

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{18 + \frac{10}{x}}{16 - \frac{5}{x}}$

Step 5.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- \infty$ .

$\frac{lim_{x \to - \infty} 18 + \frac{10}{x}}{lim_{x \to - \infty} 16 - \frac{5}{x}}$

Step 5.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \infty$ .

$\frac{lim_{x \to - \infty} 18 + lim_{x \to - \infty} \frac{10}{x}}{lim_{x \to - \infty} 16 - \frac{5}{x}}$

Step 5.5

Evaluate the limit of $18$ which is constant as $x$ approaches $- \infty$ .

$\frac{18 + lim_{x \to - \infty} \frac{10}{x}}{lim_{x \to - \infty} 16 - \frac{5}{x}}$

Step 5.6

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{18 + 10 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 16 - \frac{5}{x}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{18 + 10 \cdot 0}{lim_{x \to - \infty} 16 - \frac{5}{x}}$

Step 7

Evaluate the limit.

Tap for more steps...

Step 7.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \infty$ .

$\frac{18 + 10 \cdot 0}{lim_{x \to - \infty} 16 - lim_{x \to - \infty} \frac{5}{x}}$

Step 7.2

Evaluate the limit of $16$ which is constant as $x$ approaches $- \infty$ .

$\frac{18 + 10 \cdot 0}{16 - lim_{x \to - \infty} \frac{5}{x}}$

Step 7.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{18 + 10 \cdot 0}{16 - 5 lim_{x \to - \infty} \frac{1}{x}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{18 + 10 \cdot 0}{16 - 5 \cdot 0}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Cancel the common factor of $18 + 10 \cdot 0$ and $16 - 5 \cdot 0$ .

Tap for more steps...

Step 9.1.1

Rewrite $16$ as $- 1 (- 16)$ .

$\frac{18 + 10 \cdot 0}{- 1 (- 16) - 5 \cdot 0}$

Step 9.1.2

Factor $- 1$ out of $- 5 \cdot 0$ .

$\frac{18 + 10 \cdot 0}{- 1 (- 16) - (5 \cdot 0)}$

Step 9.1.3

Factor $- 1$ out of $- 1 (- 16) - (5 \cdot 0)$ .

$\frac{18 + 10 \cdot 0}{- 1 (- 16 + 5 \cdot 0)}$

Step 9.1.4

Reorder terms.

$\frac{18 + 10 \cdot 0}{- 1 (- 16 + 0 \cdot 5)}$

Step 9.1.5

Factor $2$ out of $18$ .

$\frac{2 (9) + 10 \cdot 0}{- 1 (- 16 + 0 \cdot 5)}$

Step 9.1.6

Factor $2$ out of $10 \cdot 0$ .

$\frac{2 (9) + 2 (5 \cdot 0)}{- 1 (- 16 + 0 \cdot 5)}$

Step 9.1.7

Factor $2$ out of $2 (9) + 2 (5 \cdot 0)$ .

$\frac{2 (9 + 5 \cdot 0)}{- 1 (- 16 + 0 \cdot 5)}$

Step 9.1.8

Cancel the common factors.

Tap for more steps...

Step 9.1.8.1

Factor $2$ out of $- 1 (- 16 + 0 \cdot 5)$ .

$\frac{2 (9 + 5 \cdot 0)}{2 (- 1 (- 8 + 0 \cdot 5))}$

Step 9.1.8.2

Cancel the common factor.

$\frac{2 (9 + 5 \cdot 0)}{2 (- 1 (- 8 + 0 \cdot 5))}$

Step 9.1.8.3

Rewrite the expression.

$\frac{9 + 5 \cdot 0}{- 1 (- 8 + 0 \cdot 5)}$

Step 9.2

Simplify the numerator.

Tap for more steps...

Step 9.2.1

Multiply $5$ by $0$ .

$\frac{9 + 0}{- 1 (- 8 + 0 \cdot 5)}$

Step 9.2.2

Add $9$ and $0$ .

$\frac{9}{- 1 (- 8 + 0 \cdot 5)}$

Step 9.3

Simplify the denominator.

Tap for more steps...

Step 9.3.1

Multiply $0$ by $5$ .

$\frac{9}{- 1 (- 8 + 0)}$

Step 9.3.2

Add $- 8$ and $0$ .

$\frac{9}{- 1 \cdot - 8}$

Step 9.4

Multiply $- 1$ by $- 8$ .

$\frac{9}{8}$