Calculus Examples

$lim_{x \to \infty} \frac{10 - 6 x^{2}}{5 + 3 e^{x}}$

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{lim_{x \to \infty} 10 - 6 x^{2}}{lim_{x \to \infty} 5 + 3 e^{x}}$

Step 1.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.1.2.1

Reorder $10$ and $- 6 x^{2}$ .

$\frac{lim_{x \to \infty} - 6 x^{2} + 10}{lim_{x \to \infty} 5 + 3 e^{x}}$

Step 1.1.2.2

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

$\frac{- \infty}{lim_{x \to \infty} 5 + 3 e^{x}}$

Step 1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.1.3.1

Evaluate the limit.

Tap for more steps...

Step 1.1.3.1.1

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

$\frac{- \infty}{lim_{x \to \infty} 5 + lim_{x \to \infty} 3 e^{x}}$

Step 1.1.3.1.2

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

$\frac{- \infty}{5 + lim_{x \to \infty} 3 e^{x}}$

Step 1.1.3.2

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

Tap for more steps...

Step 1.1.3.2.1

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

$\frac{- \infty}{5 + lim_{x \to \infty} e^{x}}$

Step 1.1.3.2.2

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

$\frac{- \infty}{5 + \infty}$

Step 1.1.3.3

Infinity plus or minus a number is infinity.

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

Step 1.1.3.4

Infinity divided by infinity is undefined.

Undefined

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

Step 1.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 1.2

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.

$lim_{x \to \infty} \frac{10 - 6 x^{2}}{5 + 3 e^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [10 - 6 x^{2}]}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [10 - 6 x^{2}]}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3.2

By the Sum Rule, the derivative of $10 - 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [- 6 x^{2}]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [10] + \frac{d}{d x} [- 6 x^{2}]}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3.3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{0 + \frac{d}{d x} [- 6 x^{2}]}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

Tap for more steps...

Step 1.3.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to \infty} \frac{0 - 6 \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3.4.2

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

$lim_{x \to \infty} \frac{0 - 6 (2 x)}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3.4.3

Multiply $2$ by $- 6$ .

$lim_{x \to \infty} \frac{0 - 12 x}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3.5

Subtract $12 x$ from $0$ .

$lim_{x \to \infty} \frac{- 12 x}{\frac{d}{d x} [5 + 3 e^{x}]}$

Step 1.3.6

By the Sum Rule, the derivative of $5 + 3 e^{x}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [3 e^{x}]$ .

$lim_{x \to \infty} \frac{- 12 x}{\frac{d}{d x} [5] + \frac{d}{d x} [3 e^{x}]}$

Step 1.3.7

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{- 12 x}{0 + \frac{d}{d x} [3 e^{x}]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [3 e^{x}]$ .

Tap for more steps...

Step 1.3.8.1

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

$lim_{x \to \infty} \frac{- 12 x}{0 + 3 \frac{d}{d x} [e^{x}]}$

Step 1.3.8.2

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

$lim_{x \to \infty} \frac{- 12 x}{0 + 3 e^{x}}$

Step 1.3.9

Add $0$ and $3 e^{x}$ .

$lim_{x \to \infty} \frac{- 12 x}{3 e^{x}}$

Step 1.4

Cancel the common factor of $- 12$ and $3$ .

Tap for more steps...

Step 1.4.1

Factor $3$ out of $- 12 x$ .

$lim_{x \to \infty} \frac{3 (- 4 x)}{3 e^{x}}$

Step 1.4.2

Cancel the common factors.

Tap for more steps...

Step 1.4.2.1

Factor $3$ out of $3 e^{x}$ .

$lim_{x \to \infty} \frac{3 (- 4 x)}{3 (e^{x})}$

Step 1.4.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{3 (- 4 x)}{3 e^{x}}$

Step 1.4.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{- 4 x}{e^{x}}$

Step 2

Move the term $- 4$ outside of the limit because it is constant with respect to $x$ .

$- 4 lim_{x \to \infty} \frac{x}{e^{x}}$

Step 3

Apply L'Hospital's rule.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

$- 4 \frac{lim_{x \to \infty} x}{lim_{x \to \infty} e^{x}}$

Step 3.1.2

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

$- 4 \frac{\infty}{lim_{x \to \infty} e^{x}}$

Step 3.1.3

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

$- 4 \frac{\infty}{\infty}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$- 4 \frac{\infty}{\infty}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.3.1

Differentiate the numerator and denominator.

$- 4 lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [e^{x}]}$

Step 3.3.2

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

$- 4 lim_{x \to \infty} \frac{1}{\frac{d}{d x} [e^{x}]}$

Step 3.3.3

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

$- 4 lim_{x \to \infty} \frac{1}{e^{x}}$

Step 4

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

$- 4 \cdot 0$

Step 5

Multiply $- 4$ by $0$ .

$0$