Calculus Examples

$lim_{t \to \infty} \frac{\sqrt{t} + t^{2}}{4 t - t^{2}}$

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.

$\frac{lim_{t \to \infty} \sqrt{t} + t^{2}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.2.1

Multiply to rationalize the numerator.

$\frac{lim_{t \to \infty} \frac{(\sqrt{t} + t^{2}) (\sqrt{t} - t^{2})}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2

Simplify.

Tap for more steps...

Step 1.2.2.1

Expand the numerator using the FOIL method.

$\frac{lim_{t \to \infty} \frac{{\sqrt{t}}^{2} + \sqrt{t} (- t^{2}) + \sqrt{t} t^{2} + t^{4} \cdot - 1}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2

Simplify.

Tap for more steps...

Step 1.2.2.2.1

Rewrite ${\sqrt{t}}^{2}$ as $t$ .

Tap for more steps...

Step 1.2.2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{lim_{t \to \infty} \frac{{(t^{\frac{1}{2}})}^{2} + t^{4} \cdot - 1}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{lim_{t \to \infty} \frac{t^{\frac{1}{2} \cdot 2} + t^{4} \cdot - 1}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{lim_{t \to \infty} \frac{t^{\frac{2}{2}} + t^{4} \cdot - 1}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.2.2.1.4.1

Cancel the common factor.

$\frac{lim_{t \to \infty} \frac{t^{\frac{2}{2}} + t^{4} \cdot - 1}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2.1.4.2

Rewrite the expression.

$\frac{lim_{t \to \infty} \frac{t^{1} + t^{4} \cdot - 1}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2.1.5

Simplify.

$\frac{lim_{t \to \infty} \frac{t + t^{4} \cdot - 1}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2.2

Move $- 1$ to the left of $t^{4}$ .

$\frac{lim_{t \to \infty} \frac{t - 1 \cdot t^{4}}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.2.2.3

Rewrite $- 1 t^{4}$ as $- t^{4}$ .

$\frac{lim_{t \to \infty} \frac{t - t^{4}}{\sqrt{t} - t^{2}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.3

Divide the numerator and denominator by the highest power of $t$ in the denominator, which is $t^{2} = \sqrt{t^{4}}$ .

$\frac{lim_{t \to \infty} \frac{\frac{t}{t^{2}} - \frac{t^{4}}{t^{2}}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4

Simplify terms.

Tap for more steps...

Step 1.2.4.1

Simplify each term.

Tap for more steps...

Step 1.2.4.1.1

Cancel the common factor of $t$ and $t^{2}$ .

Tap for more steps...

Step 1.2.4.1.1.1

Raise $t$ to the power of $1$ .

$\frac{lim_{t \to \infty} \frac{\frac{t^{1}}{t^{2}} - \frac{t^{4}}{t^{2}}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.1.2

Factor $t$ out of $t^{1}$ .

$\frac{lim_{t \to \infty} \frac{\frac{t \cdot 1}{t^{2}} - \frac{t^{4}}{t^{2}}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.1.3

Cancel the common factors.

Tap for more steps...

Step 1.2.4.1.1.3.1

Factor $t$ out of $t^{2}$ .

$\frac{lim_{t \to \infty} \frac{\frac{t \cdot 1}{t \cdot t} - \frac{t^{4}}{t^{2}}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.1.3.2

Cancel the common factor.

$\frac{lim_{t \to \infty} \frac{\frac{t \cdot 1}{t \cdot t} - \frac{t^{4}}{t^{2}}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.1.3.3

Rewrite the expression.

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - \frac{t^{4}}{t^{2}}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.2

Cancel the common factor of $t^{4}$ and $t^{2}$ .

Tap for more steps...

Step 1.2.4.1.2.1

Factor $t^{2}$ out of $t^{4}$ .

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - \frac{t^{2} t^{2}}{t^{2}}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.1.2.2.1

Multiply by $1$ .

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - \frac{t^{2} t^{2}}{t^{2} \cdot 1}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.2.2.2

Cancel the common factor.

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - \frac{t^{2} t^{2}}{t^{2} \cdot 1}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.2.2.3

Rewrite the expression.

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - \frac{t^{2}}{1}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.1.2.2.4

Divide $t^{2}$ by $1$ .

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - t^{2}}{\sqrt{\frac{t}{t^{4}}} - \frac{t^{2}}{t^{2}}}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.2

Simplify each term.

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - t^{2}}{\sqrt{\frac{t}{t^{4}}} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.3

Cancel the common factor of $t$ and $t^{4}$ .

Tap for more steps...

Step 1.2.4.3.1

Raise $t$ to the power of $1$ .

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - t^{2}}{\sqrt{\frac{t^{1}}{t^{4}}} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.3.2

Factor $t$ out of $t^{1}$ .

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - t^{2}}{\sqrt{\frac{t \cdot 1}{t^{4}}} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.3.3

Cancel the common factors.

Tap for more steps...

Step 1.2.4.3.3.1

Factor $t$ out of $t^{4}$ .

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - t^{2}}{\sqrt{\frac{t \cdot 1}{t \cdot t^{3}}} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.3.3.2

Cancel the common factor.

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - t^{2}}{\sqrt{\frac{t \cdot 1}{t \cdot t^{3}}} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.4.3.3.3

Rewrite the expression.

$\frac{lim_{t \to \infty} \frac{\frac{1}{t} - t^{2}}{\sqrt{\frac{1}{t^{3}}} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.5

As $t$ approaches $\infty$ , the fraction $\frac{1}{t}$ approaches $0$ .

$\frac{lim_{t \to \infty} \frac{0 - t^{2}}{\sqrt{\frac{1}{t^{3}}} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.6

As $t$ approaches $\infty$ , the fraction $\frac{1}{t^{3}}$ approaches $0$ .

$\frac{lim_{t \to \infty} \frac{0 - t^{2}}{\sqrt{0} - 1}}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.2.7

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{0 - t^{2}}{\sqrt{0} - 1}$ approaches infinity.

$\frac{\infty}{lim_{t \to \infty} 4 t - t^{2}}$

Step 1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.3.1

Reorder $4 t$ and $- t^{2}$ .

$\frac{\infty}{lim_{t \to \infty} - t^{2} + 4 t}$

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.

$lim_{t \to \infty} \frac{\sqrt{t} + t^{2}}{4 t - t^{2}} = lim_{t \to \infty} \frac{\frac{d}{d t} [\sqrt{t} + t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.1

Differentiate the numerator and denominator.

$lim_{t \to \infty} \frac{\frac{d}{d t} [\sqrt{t} + t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.2

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

$lim_{t \to \infty} \frac{\frac{d}{d t} [\sqrt{t}] + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3

Evaluate $\frac{d}{d t} [\sqrt{t}]$ .

Tap for more steps...

Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$lim_{t \to \infty} \frac{\frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3.2

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

$lim_{t \to \infty} \frac{\frac{1}{2} t^{\frac{1}{2} - 1} + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{t \to \infty} \frac{\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3.4

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{t \to \infty} \frac{\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3.5

Combine the numerators over the common denominator.

$lim_{t \to \infty} \frac{\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3.6

Simplify the numerator.

Tap for more steps...

Step 3.3.6.1

Multiply $- 1$ by $2$ .

$lim_{t \to \infty} \frac{\frac{1}{2} t^{\frac{1 - 2}{2}} + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3.6.2

Subtract $2$ from $1$ .

$lim_{t \to \infty} \frac{\frac{1}{2} t^{\frac{- 1}{2}} + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.3.7

Move the negative in front of the fraction.

$lim_{t \to \infty} \frac{\frac{1}{2} t^{- \frac{1}{2}} + \frac{d}{d t} [t^{2}]}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.4

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

$lim_{t \to \infty} \frac{\frac{1}{2} t^{- \frac{1}{2}} + 2 t}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{t \to \infty} \frac{\frac{1}{2} \cdot \frac{1}{t^{\frac{1}{2}}} + 2 t}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.5.2

Multiply $\frac{1}{2}$ by $\frac{1}{t^{\frac{1}{2}}}$ .

$lim_{t \to \infty} \frac{\frac{1}{2 t^{\frac{1}{2}}} + 2 t}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.5.3

Reorder terms.

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{\frac{d}{d t} [4 t - t^{2}]}$

Step 3.6

By the Sum Rule, the derivative of $4 t - t^{2}$ with respect to $t$ is $\frac{d}{d t} [4 t] + \frac{d}{d t} [- t^{2}]$ .

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{\frac{d}{d t} [4 t] + \frac{d}{d t} [- t^{2}]}$

Step 3.7

Evaluate $\frac{d}{d t} [4 t]$ .

Tap for more steps...

Step 3.7.1

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{4 \frac{d}{d t} [t] + \frac{d}{d t} [- t^{2}]}$

Step 3.7.2

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

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{4 \cdot 1 + \frac{d}{d t} [- t^{2}]}$

Step 3.7.3

Multiply $4$ by $1$ .

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{4 + \frac{d}{d t} [- t^{2}]}$

Step 3.8

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

Tap for more steps...

Step 3.8.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- t^{2}$ with respect to $t$ is $- \frac{d}{d t} [t^{2}]$ .

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{4 - \frac{d}{d t} [t^{2}]}$

Step 3.8.2

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

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{4 - (2 t)}$

Step 3.8.3

Multiply $2$ by $- 1$ .

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{4 - 2 t}$

Step 3.9

Reorder terms.

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 t^{\frac{1}{2}}}}{- 2 t + 4}$

Step 4

Rewrite $t^{\frac{1}{2}}$ as $\sqrt{t}$ .

$lim_{t \to \infty} \frac{2 t + \frac{1}{2 \sqrt{t}}}{- 2 t + 4}$

Step 5

Combine terms.

Tap for more steps...

Step 5.1

To write $2 t$ as a fraction with a common denominator, multiply by $\frac{2 \sqrt{t}}{2 \sqrt{t}}$ .

$lim_{t \to \infty} \frac{2 t \cdot \frac{2 \sqrt{t}}{2 \sqrt{t}} + \frac{1}{2 \sqrt{t}}}{- 2 t + 4}$

Step 5.2

Combine $2 t$ and $\frac{2 \sqrt{t}}{2 \sqrt{t}}$ .

$lim_{t \to \infty} \frac{\frac{2 t (2 \sqrt{t})}{2 \sqrt{t}} + \frac{1}{2 \sqrt{t}}}{- 2 t + 4}$

Step 5.3

Combine the numerators over the common denominator.

$lim_{t \to \infty} \frac{\frac{2 t (2 \sqrt{t}) + 1}{2 \sqrt{t}}}{- 2 t + 4}$