Calculus Examples

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

Step 1

Evaluate the limit of $\frac{\sqrt{t} + t^{2}}{2 t - t^{2}}$ which is constant as $x$ approaches $\infty$ .

$\frac{\sqrt{t} + t^{2}}{2 t - t^{2}}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

$\frac{t^{\frac{1}{2}} + t^{2}}{2 t - t^{2}}$

Step 2.1.2

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} + t^{2}$ .

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Step 2.1.2.1

Multiply by $1$ .

$\frac{t^{\frac{1}{2}} \cdot 1 + t^{2}}{2 t - t^{2}}$

Step 2.1.2.2

Factor $t^{\frac{1}{2}}$ out of $t^{2}$ .

$\frac{t^{\frac{1}{2}} \cdot 1 + t^{\frac{1}{2}} t^{\frac{3}{2}}}{2 t - t^{2}}$

Step 2.1.2.3

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} \cdot 1 + t^{\frac{1}{2}} t^{\frac{3}{2}}$ .

$\frac{t^{\frac{1}{2}} (1 + t^{\frac{3}{2}})}{2 t - t^{2}}$

Step 2.1.3

Rewrite $1$ as $1^{3}$ .

$\frac{t^{\frac{1}{2}} (1^{3} + t^{\frac{3}{2}})}{2 t - t^{2}}$

Step 2.1.4

Rewrite $t^{\frac{3}{2}}$ as ${(t^{\frac{1}{2}})}^{3}$ .

$\frac{t^{\frac{1}{2}} (1^{3} + {(t^{\frac{1}{2}})}^{3})}{2 t - t^{2}}$

Step 2.1.5

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = t^{\frac{1}{2}}$ .

$\frac{t^{\frac{1}{2}} ((1 + t^{\frac{1}{2}}) (1^{2} - 1 t^{\frac{1}{2}} + {(t^{\frac{1}{2}})}^{2}))}{2 t - t^{2}}$

Step 2.1.6

Simplify.

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Step 2.1.6.1

One to any power is one.

$\frac{t^{\frac{1}{2}} ((1 + t^{\frac{1}{2}}) (1 - 1 t^{\frac{1}{2}} + {(t^{\frac{1}{2}})}^{2}))}{2 t - t^{2}}$

Step 2.1.6.2

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

$\frac{t^{\frac{1}{2}} ((1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + {(t^{\frac{1}{2}})}^{2}))}{2 t - t^{2}}$

Step 2.1.6.3

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 2.1.6.3.1

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

$\frac{t^{\frac{1}{2}} ((1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t^{\frac{1}{2} \cdot 2}))}{2 t - t^{2}}$

Step 2.1.6.3.2

Cancel the common factor of $2$ .

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Step 2.1.6.3.2.1

Cancel the common factor.

$\frac{t^{\frac{1}{2}} ((1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t^{\frac{1}{2} \cdot 2}))}{2 t - t^{2}}$

Step 2.1.6.3.2.2

Rewrite the expression.

$\frac{t^{\frac{1}{2}} ((1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t^{1}))}{2 t - t^{2}}$

Step 2.1.6.4

Simplify.

$\frac{t^{\frac{1}{2}} ((1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t))}{2 t - t^{2}}$

$\frac{t^{\frac{1}{2}} (1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{2 t - t^{2}}$

Step 2.2

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

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Step 2.2.1

Factor $t$ out of $2 t$ .

$\frac{t^{\frac{1}{2}} (1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t \cdot 2 - t^{2}}$

Step 2.2.2

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

$\frac{t^{\frac{1}{2}} (1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t \cdot 2 + t (- t)}$

Step 2.2.3

Factor $t$ out of $t \cdot 2 + t (- t)$ .

$\frac{t^{\frac{1}{2}} (1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t (2 - t)}$

Step 2.3

Move $t^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{(1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t (2 - t) t^{- \frac{1}{2}}}$

Step 2.4

Multiply $t$ by $t^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.4.1

Move $t^{- \frac{1}{2}}$ .

$\frac{(1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t^{- \frac{1}{2}} t (2 - t)}$

Step 2.4.2

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

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Step 2.4.2.1

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

$\frac{(1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t^{- \frac{1}{2}} t^{1} (2 - t)}$

Step 2.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t^{- \frac{1}{2} + 1} (2 - t)}$

Step 2.4.3

Write $1$ as a fraction with a common denominator.

$\frac{(1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t^{- \frac{1}{2} + \frac{2}{2}} (2 - t)}$

Step 2.4.4

Combine the numerators over the common denominator.

$\frac{(1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t^{\frac{- 1 + 2}{2}} (2 - t)}$

Step 2.4.5

Add $- 1$ and $2$ .

$\frac{(1 + t^{\frac{1}{2}}) (1 - t^{\frac{1}{2}} + t)}{t^{\frac{1}{2}} (2 - t)}$