Calculus Examples

$lim_{y \to 2} \frac{y + 2}{y^{2} + 5 y + 6}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $2$ .

$\frac{lim_{y \to 2} y + 2}{lim_{y \to 2} y^{2} + 5 y + 6}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $2$ .

$\frac{lim_{y \to 2} y + lim_{y \to 2} 2}{lim_{y \to 2} y^{2} + 5 y + 6}$

Step 3

Evaluate the limit of $2$ which is constant as $y$ approaches $2$ .

$\frac{lim_{y \to 2} y + 2}{lim_{y \to 2} y^{2} + 5 y + 6}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $2$ .

$\frac{lim_{y \to 2} y + 2}{lim_{y \to 2} y^{2} + lim_{y \to 2} 5 y + lim_{y \to 2} 6}$

Step 5

Move the exponent $2$ from $y^{2}$ outside the limit using the Limits Power Rule.

$\frac{lim_{y \to 2} y + 2}{{(lim_{y \to 2} y)}^{2} + lim_{y \to 2} 5 y + lim_{y \to 2} 6}$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $y$ .

$\frac{lim_{y \to 2} y + 2}{{(lim_{y \to 2} y)}^{2} + 5 lim_{y \to 2} y + lim_{y \to 2} 6}$

Step 7

Evaluate the limit of $6$ which is constant as $y$ approaches $2$ .

$\frac{lim_{y \to 2} y + 2}{{(lim_{y \to 2} y)}^{2} + 5 lim_{y \to 2} y + 6}$

Step 8

Evaluate the limits by plugging in $2$ for all occurrences of $y$ .

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Step 8.1

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

$\frac{2 + 2}{{(lim_{y \to 2} y)}^{2} + 5 lim_{y \to 2} y + 6}$

Step 8.2

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

$\frac{2 + 2}{2^{2} + 5 lim_{y \to 2} y + 6}$

Step 8.3

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

$\frac{2 + 2}{2^{2} + 5 \cdot 2 + 6}$

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $2 + 2$ and $2^{2} + 5 \cdot 2 + 6$ .

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Step 9.1.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 + 2}{2^{2} + 5 \cdot 2 + 6}$

Step 9.1.2

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 + 2 \cdot 1}{2^{2} + 5 \cdot 2 + 6}$

Step 9.1.3

Factor $2$ out of $2 \cdot 1 + 2 \cdot 1$ .

$\frac{2 \cdot (1 + 1)}{2^{2} + 5 \cdot 2 + 6}$

Step 9.1.4

Cancel the common factors.

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Step 9.1.4.1

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

$\frac{2 \cdot (1 + 1)}{2 \cdot 2 + 5 \cdot 2 + 6}$

Step 9.1.4.2

Factor $2$ out of $5 \cdot 2$ .

$\frac{2 \cdot (1 + 1)}{2 \cdot 2 + 2 \cdot 5 + 6}$

Step 9.1.4.3

Factor $2$ out of $2 \cdot 2 + 2 \cdot 5$ .

$\frac{2 \cdot (1 + 1)}{2 \cdot (2 + 5) + 6}$

Step 9.1.4.4

Factor $2$ out of $6$ .

$\frac{2 \cdot (1 + 1)}{2 \cdot (2 + 5) + 2 (3)}$

Step 9.1.4.5

Factor $2$ out of $2 \cdot (2 + 5) + 2 (3)$ .

$\frac{2 \cdot (1 + 1)}{2 \cdot (2 + 5 + 3)}$

Step 9.1.4.6

Cancel the common factor.

$\frac{2 \cdot (1 + 1)}{2 \cdot (2 + 5 + 3)}$

Step 9.1.4.7

Rewrite the expression.

$\frac{1 + 1}{2 + 5 + 3}$

Step 9.2

Add $1$ and $1$ .

$\frac{2}{2 + 5 + 3}$

Step 9.3

Simplify the denominator.

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Step 9.3.1

Add $2$ and $5$ .

$\frac{2}{7 + 3}$

Step 9.3.2

Add $7$ and $3$ .

$\frac{2}{10}$

Step 9.4

Cancel the common factor of $2$ and $10$ .

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Step 9.4.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{10}$

Step 9.4.2

Cancel the common factors.

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Step 9.4.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 1}{2 \cdot 5}$

Step 9.4.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 5}$

Step 9.4.2.3

Rewrite the expression.

$\frac{1}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{5}$

Decimal Form:

$0.2$