Calculus Examples

$lim_{x \to 6^{-}} \frac{x + 9}{x + 6}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6^{-}} x + 9}{lim_{x \to 6^{-}} x + 6}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6^{-}} x + lim_{x \to 6^{-}} 9}{lim_{x \to 6^{-}} x + 6}$

Step 3

Evaluate the limit of $9$ which is constant as $x$ approaches $6$ .

$\frac{lim_{x \to 6^{-}} x + 9}{lim_{x \to 6^{-}} x + 6}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6^{-}} x + 9}{lim_{x \to 6^{-}} x + lim_{x \to 6^{-}} 6}$

Step 5

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

$\frac{lim_{x \to 6^{-}} x + 9}{lim_{x \to 6^{-}} x + 6}$

Step 6

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6 + 9}{lim_{x \to 6^{-}} x + 6}$

Step 6.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6 + 9}{6 + 6}$

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $6 + 9$ and $6 + 6$ .

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Step 7.1.1

Factor $3$ out of $6$ .

$\frac{3 (2) + 9}{6 + 6}$

Step 7.1.2

Factor $3$ out of $9$ .

$\frac{3 \cdot 2 + 3 \cdot 3}{6 + 6}$

Step 7.1.3

Factor $3$ out of $3 \cdot 2 + 3 \cdot 3$ .

$\frac{3 \cdot (2 + 3)}{6 + 6}$

Step 7.1.4

Cancel the common factors.

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Step 7.1.4.1

Factor $3$ out of $6$ .

$\frac{3 \cdot (2 + 3)}{3 (2) + 6}$

Step 7.1.4.2

Factor $3$ out of $6$ .

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

Step 7.1.4.3

Factor $3$ out of $3 \cdot 2 + 3 \cdot 2$ .

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

Step 7.1.4.4

Cancel the common factor.

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

Step 7.1.4.5

Rewrite the expression.

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

Step 7.2

Add $2$ and $3$ .

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

Step 7.3

Add $2$ and $2$ .

$\frac{5}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{4}$

Decimal Form:

$1.25$