Calculus Examples

$lim_{x \to 5} (\sqrt{5 x} + 3 x) (x^{2} - 4)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $5$ .

$lim_{x \to 5} \sqrt{5 x} + 3 x \cdot lim_{x \to 5} x^{2} - 4$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$(lim_{x \to 5} \sqrt{5 x} + lim_{x \to 5} 3 x) \cdot lim_{x \to 5} x^{2} - 4$

Step 3

Move the limit under the radical sign.

$(\sqrt{lim_{x \to 5} 5 x} + lim_{x \to 5} 3 x) \cdot lim_{x \to 5} x^{2} - 4$

Step 4

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

$(\sqrt{5 lim_{x \to 5} x} + lim_{x \to 5} 3 x) \cdot lim_{x \to 5} x^{2} - 4$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$(\sqrt{5 lim_{x \to 5} x} + 3 lim_{x \to 5} x) \cdot lim_{x \to 5} x^{2} - 4$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$(\sqrt{5 lim_{x \to 5} x} + 3 lim_{x \to 5} x) \cdot (lim_{x \to 5} x^{2} - lim_{x \to 5} 4)$

Step 7

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

$(\sqrt{5 lim_{x \to 5} x} + 3 lim_{x \to 5} x) \cdot ({(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 4)$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $5$ .

$(\sqrt{5 lim_{x \to 5} x} + 3 lim_{x \to 5} x) \cdot ({(lim_{x \to 5} x)}^{2} - 1 \cdot 4)$

Step 9

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$(\sqrt{5 \cdot 5} + 3 lim_{x \to 5} x) \cdot ({(lim_{x \to 5} x)}^{2} - 1 \cdot 4)$

Step 9.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$(\sqrt{5 \cdot 5} + 3 \cdot 5) \cdot ({(lim_{x \to 5} x)}^{2} - 1 \cdot 4)$

Step 9.3

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$(\sqrt{5 \cdot 5} + 3 \cdot 5) \cdot (5^{2} - 1 \cdot 4)$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply $5$ by $5$ .

$(\sqrt{25} + 3 \cdot 5) \cdot (5^{2} - 1 \cdot 4)$

Step 10.1.2

Rewrite $25$ as $5^{2}$ .

$(\sqrt{5^{2}} + 3 \cdot 5) \cdot (5^{2} - 1 \cdot 4)$

Step 10.1.3

Pull terms out from under the radical, assuming positive real numbers.

$(5 + 3 \cdot 5) \cdot (5^{2} - 1 \cdot 4)$

Step 10.1.4

Multiply $3$ by $5$ .

$(5 + 15) \cdot (5^{2} - 1 \cdot 4)$

Step 10.2

Add $5$ and $15$ .

$20 \cdot (5^{2} - 1 \cdot 4)$

Step 10.3

Simplify each term.

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Step 10.3.1

Raise $5$ to the power of $2$ .

$20 \cdot (25 - 1 \cdot 4)$

Step 10.3.2

Multiply $- 1$ by $4$ .

$20 \cdot (25 - 4)$

Step 10.4

Subtract $4$ from $25$ .

$20 \cdot 21$

Step 10.5

Multiply $20$ by $21$ .

$420$