Calculus Examples

$lim_{x \to 0} x \cos (x^{2} + \frac{1}{x})$

Step 1

Combine terms.

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Step 1.1

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

$lim_{x \to 0} x \cos (\frac{x^{2} x}{x} + \frac{1}{x})$

Step 1.2

Combine the numerators over the common denominator.

$lim_{x \to 0} x \cos (\frac{x^{2} x + 1}{x})$

Step 2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1

Multiply $x^{2}$ by $x$ .

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Step 2.1.1

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

$lim_{x \to 0} x \cos (\frac{x^{2} x^{1} + 1}{x})$

Step 2.1.2

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

$lim_{x \to 0} x \cos (\frac{x^{2 + 1} + 1}{x})$

Step 2.2

Add $2$ and $1$ .

$lim_{x \to 0} x \cos (\frac{x^{3} + 1}{x})$

Step 3

Since $x \cdot - 1 \leq x \cos (\frac{x^{3} + 1}{x}) \leq x \cdot 1$ and $lim_{x \to 0} x \cdot - 1 = lim_{x \to 0} x \cdot 1 = 0$ , apply the squeeze theorem.

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