Calculus Examples

$lim_{x \to - 3} (9 - 7 x) \sqrt{6 - 4 x}$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 3$ .

$lim_{x \to - 3} 9 - 7 x \cdot lim_{x \to - 3} \sqrt{6 - 4 x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$(lim_{x \to - 3} 9 - lim_{x \to - 3} 7 x) \cdot lim_{x \to - 3} \sqrt{6 - 4 x}$

Step 3

Evaluate the limit of $9$ which is constant as $x$ approaches $- 3$ .

$(9 - lim_{x \to - 3} 7 x) \cdot lim_{x \to - 3} \sqrt{6 - 4 x}$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$(9 - 7 lim_{x \to - 3} x) \cdot lim_{x \to - 3} \sqrt{6 - 4 x}$

Step 5

Move the limit under the radical sign.

$(9 - 7 lim_{x \to - 3} x) \cdot \sqrt{lim_{x \to - 3} 6 - 4 x}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$(9 - 7 lim_{x \to - 3} x) \cdot \sqrt{lim_{x \to - 3} 6 - lim_{x \to - 3} 4 x}$

Step 7

Evaluate the limit of $6$ which is constant as $x$ approaches $- 3$ .

$(9 - 7 lim_{x \to - 3} x) \cdot \sqrt{6 - lim_{x \to - 3} 4 x}$

Step 8

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$(9 - 7 lim_{x \to - 3} x) \cdot \sqrt{6 - 4 lim_{x \to - 3} x}$

Step 9

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$(9 - 7 \cdot - 3) \cdot \sqrt{6 - 4 lim_{x \to - 3} x}$

Step 9.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$(9 - 7 \cdot - 3) \cdot \sqrt{6 - 4 \cdot - 3}$

Step 10

Simplify the answer.

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Step 10.1

Multiply $- 7$ by $- 3$ .

$(9 + 21) \cdot \sqrt{6 - 4 \cdot - 3}$

Step 10.2

Add $9$ and $21$ .

$30 \cdot \sqrt{6 - 4 \cdot - 3}$

Step 10.3

Multiply $- 4$ by $- 3$ .

$30 \cdot \sqrt{6 + 12}$

Step 10.4

Add $6$ and $12$ .

$30 \cdot \sqrt{18}$

Step 10.5

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 10.5.1

Factor $9$ out of $18$ .

$30 \cdot \sqrt{9 (2)}$

Step 10.5.2

Rewrite $9$ as $3^{2}$ .

$30 \cdot \sqrt{3^{2} \cdot 2}$

Step 10.6

Pull terms out from under the radical.

$30 \cdot (3 \sqrt{2})$

Step 10.7

Multiply $3$ by $30$ .

$90 \sqrt{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$90 \sqrt{2}$

Decimal Form:

$127.27922061 \dots$