Calculus Examples

$lim_{x \to 0^{+}} \frac{1}{t \sqrt{1 + t}} - \frac{1}{t}$

Step 1

Evaluate the limit of $\frac{1}{t \sqrt{1 + t}} - \frac{1}{t}$ which is constant as $x$ approaches $0$ .

$\frac{1}{t \sqrt{1 + t}} - \frac{1}{t}$

Step 2

Simplify the answer.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\frac{1}{t \sqrt{1 + t}}$ by $\frac{\sqrt{1 + t}}{\sqrt{1 + t}}$ .

$\frac{1}{t \sqrt{1 + t}} \cdot \frac{\sqrt{1 + t}}{\sqrt{1 + t}} - \frac{1}{t}$

Step 2.1.2

Combine and simplify the denominator.

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Step 2.1.2.1

Multiply $\frac{1}{t \sqrt{1 + t}}$ by $\frac{\sqrt{1 + t}}{\sqrt{1 + t}}$ .

$\frac{\sqrt{1 + t}}{t \sqrt{1 + t} \sqrt{1 + t}} - \frac{1}{t}$

Step 2.1.2.2

Move $\sqrt{1 + t}$ .

$\frac{\sqrt{1 + t}}{t (\sqrt{1 + t} \sqrt{1 + t})} - \frac{1}{t}$

Step 2.1.2.3

Raise $\sqrt{1 + t}$ to the power of $1$ .

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

Step 2.1.2.4

Raise $\sqrt{1 + t}$ to the power of $1$ .

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

Step 2.1.2.5

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

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

Step 2.1.2.6

Add $1$ and $1$ .

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

Step 2.1.2.7

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

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Step 2.1.2.7.1

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

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

Step 2.1.2.7.2

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

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

Step 2.1.2.7.3

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

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

Step 2.1.2.7.4

Cancel the common factor of $2$ .

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Step 2.1.2.7.4.1

Cancel the common factor.

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

Step 2.1.2.7.4.2

Rewrite the expression.

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

Step 2.1.2.7.5

Simplify.

$\frac{\sqrt{1 + t}}{t (1 + t)} - \frac{1}{t}$

Step 2.2

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

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

Step 2.3

Multiply $\frac{1}{t}$ by $\frac{1 + t}{1 + t}$ .

$\frac{\sqrt{1 + t}}{t (1 + t)} - \frac{1 + t}{t (1 + t)}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\sqrt{1 + t} - (1 + t)}{t (1 + t)}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

Let $u = {(1 + t)}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${(1 + t)}^{\frac{1}{2}}$ .

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

Step 2.5.4

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

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Step 2.5.4.1

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

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

Step 2.5.4.2

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

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

Step 2.5.4.3

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

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

Step 2.5.4.4

Factor $u$ out of $u \cdot 1 + u (- u)$ .

$\frac{u (1 - u)}{t (1 + t)}$

Step 2.5.5

Replace all occurrences of $u$ with ${(1 + t)}^{\frac{1}{2}}$ .

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

Step 2.6

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

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

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

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Step 2.7.2.1

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

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

Step 2.7.2.2

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

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

Step 2.7.3

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

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

Step 2.7.4

Combine the numerators over the common denominator.

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

Step 2.7.5

Add $- 1$ and $2$ .

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