Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

Move $t^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$lim_{x \to \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} {(t^{\frac{1}{2}})}^{- 1} d t$

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Move the negative in front of the fraction.

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

Step 2

By the Power Rule, the integral of $t^{- \frac{1}{2}}$ with respect to $t$ is $2 t^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{1}{\sqrt{x}} 2 t^{\frac{1}{2}}]_{1}^{x}$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{\sqrt{x}}$ and $2 t^{\frac{1}{2}}]_{1}^{x}$ .

$lim_{x \to \infty} \frac{2 t^{\frac{1}{2}}]_{1}^{x}}{\sqrt{x}}$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $2 t^{\frac{1}{2}}$ at $x$ and at $1$ .

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

One to any power is one.

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

Step 3.2.2.2

Multiply $- 2$ by $1$ .

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