Calculus Examples

$lim_{y \to 0} \frac{x^{2} + y}{x + 2 y}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $0$ .

$\frac{lim_{y \to 0} x^{2} + y}{lim_{y \to 0} x + 2 y}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $0$ .

$\frac{lim_{y \to 0} x^{2} + lim_{y \to 0} y}{lim_{y \to 0} x + 2 y}$

Step 3

Evaluate the limit of $x^{2}$ which is constant as $y$ approaches $0$ .

$\frac{x^{2} + lim_{y \to 0} y}{lim_{y \to 0} x + 2 y}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $0$ .

$\frac{x^{2} + lim_{y \to 0} y}{lim_{y \to 0} x + lim_{y \to 0} 2 y}$

Step 5

Evaluate the limit of $x$ which is constant as $y$ approaches $0$ .

$\frac{x^{2} + lim_{y \to 0} y}{x + lim_{y \to 0} 2 y}$

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $y$ .

$\frac{x^{2} + lim_{y \to 0} y}{x + 2 lim_{y \to 0} y}$

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $y$ .

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Step 7.1

Evaluate the limit of $y$ by plugging in $0$ for $y$ .

$\frac{x^{2} + 0}{x + 2 lim_{y \to 0} y}$

Step 7.2

Evaluate the limit of $y$ by plugging in $0$ for $y$ .

$\frac{x^{2} + 0}{x + 2 \cdot 0}$

Step 8

Simplify the answer.

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Step 8.1

Add $x^{2}$ and $0$ .

$\frac{x^{2}}{x + 2 \cdot 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $0$ .

$\frac{x^{2}}{x + 0}$

Step 8.2.2

Add $x$ and $0$ .

$\frac{x^{2}}{x}$

Step 8.3

Cancel the common factor of $x^{2}$ and $x$ .

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Step 8.3.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x}{x}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

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

$\frac{x \cdot x}{x^{1}}$

Step 8.3.2.2

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

$\frac{x \cdot x}{x \cdot 1}$

Step 8.3.2.3

Cancel the common factor.

$\frac{x \cdot x}{x \cdot 1}$

Step 8.3.2.4

Rewrite the expression.

$\frac{x}{1}$

Step 8.3.2.5

Divide $x$ by $1$ .

$x$