Calculus Examples

$lim_{x \to \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} \frac{1}{\sqrt{t}} d t$

Step 1

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

$\frac{d}{d x} [lim_{x \to \infty} \frac{1}{x^{\frac{1}{2}}} \int_{1}^{x} \frac{1}{\sqrt{t}} d t]$

Step 2

Simplify.

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Step 2.1

Move the term $\int_{1}^{x} \frac{1}{\sqrt{t}} d t$ outside of the limit because it is constant with respect to $x$ .

$\frac{d}{d x} [\int_{1}^{x} \frac{1}{\sqrt{t}} d t lim_{x \to \infty} \frac{1}{x^{\frac{1}{2}}}]$

Step 2.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{\frac{1}{2}}}$ approaches $0$ .

$\frac{d}{d x} [\int_{1}^{x} \frac{1}{\sqrt{t}} d t \cdot 0]$

Step 2.3

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

$\frac{d}{d x} [\int_{1}^{x} \frac{1}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\sqrt{t}} d t \cdot 0]$

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

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

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{\sqrt{t} \sqrt{t}} d t \cdot 0]$

Step 2.4.2

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

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{{\sqrt{t}}^{1} \sqrt{t}} d t \cdot 0]$

Step 2.4.3

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

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{{\sqrt{t}}^{1} {\sqrt{t}}^{1}} d t \cdot 0]$

Step 2.4.4

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

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{{\sqrt{t}}^{1 + 1}} d t \cdot 0]$

Step 2.4.5

Add $1$ and $1$ .

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{{\sqrt{t}}^{2}} d t \cdot 0]$

Step 2.4.6

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

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Step 2.4.6.1

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

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{{(t^{\frac{1}{2}})}^{2}} d t \cdot 0]$

Step 2.4.6.2

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

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{t^{\frac{1}{2} \cdot 2}} d t \cdot 0]$

Step 2.4.6.3

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

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{t^{\frac{2}{2}}} d t \cdot 0]$

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{t^{\frac{2}{2}}} d t \cdot 0]$

Step 2.4.6.4.2

Rewrite the expression.

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{t^{1}} d t \cdot 0]$

Step 2.4.6.5

Simplify.

$\frac{d}{d x} [\int_{1}^{x} \frac{\sqrt{t}}{t} d t \cdot 0]$

Step 2.5

Multiply $\int_{1}^{x} \frac{\sqrt{t}}{t} d t$ by $0$ .

$\frac{d}{d x} [0]$