Calculus Examples

$lim_{x \to 8} \frac{6000 x^{2}}{2 x^{2} + 3000}$

Step 1

Cancel the common factor of $6000$ and $2 x^{2} + 3000$ .

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Step 1.1

Factor $2$ out of $6000 x^{2}$ .

$lim_{x \to 8} \frac{2 (3000 x^{2})}{2 x^{2} + 3000}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

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

$lim_{x \to 8} \frac{2 (3000 x^{2})}{2 (x^{2}) + 3000}$

Step 1.2.2

Factor $2$ out of $3000$ .

$lim_{x \to 8} \frac{2 (3000 x^{2})}{2 x^{2} + 2 \cdot 1500}$

Step 1.2.3

Factor $2$ out of $2 x^{2} + 2 \cdot 1500$ .

$lim_{x \to 8} \frac{2 (3000 x^{2})}{2 (x^{2} + 1500)}$

Step 1.2.4

Cancel the common factor.

$lim_{x \to 8} \frac{2 (3000 x^{2})}{2 (x^{2} + 1500)}$

Step 1.2.5

Rewrite the expression.

$lim_{x \to 8} \frac{3000 x^{2}}{x^{2} + 1500}$

Step 2

Move the term $3000$ outside of the limit because it is constant with respect to $x$ .

$3000 lim_{x \to 8} \frac{x^{2}}{x^{2} + 1500}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$3000 \frac{lim_{x \to 8} x^{2}}{lim_{x \to 8} x^{2} + 1500}$

Step 4

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

$3000 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} x^{2} + 1500}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$3000 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} x^{2} + lim_{x \to 8} 1500}$

Step 6

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

$3000 \frac{{(lim_{x \to 8} x)}^{2}}{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 1500}$

Step 7

Evaluate the limit of $1500$ which is constant as $x$ approaches $8$ .

$3000 \frac{{(lim_{x \to 8} x)}^{2}}{{(lim_{x \to 8} x)}^{2} + 1500}$

Step 8

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3000 \frac{8^{2}}{{(lim_{x \to 8} x)}^{2} + 1500}$

Step 8.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3000 \frac{8^{2}}{8^{2} + 1500}$

Step 9

Simplify the answer.

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Step 9.1

Raise $8$ to the power of $2$ .

$3000 \frac{64}{8^{2} + 1500}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Raise $8$ to the power of $2$ .

$3000 \frac{64}{64 + 1500}$

Step 9.2.2

Add $64$ and $1500$ .

$3000 (\frac{64}{1564})$

Step 9.3

Cancel the common factor of $4$ .

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Step 9.3.1

Factor $4$ out of $3000$ .

$4 (750) \frac{64}{1564}$

Step 9.3.2

Factor $4$ out of $1564$ .

$4 \cdot 750 \frac{64}{4 \cdot 391}$

Step 9.3.3

Cancel the common factor.

$4 \cdot 750 \frac{64}{4 \cdot 391}$

Step 9.3.4

Rewrite the expression.

$750 (\frac{64}{391})$

Step 9.4

Combine $750$ and $\frac{64}{391}$ .

$\frac{750 \cdot 64}{391}$

Step 9.5

Multiply $750$ by $64$ .

$\frac{48000}{391}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{48000}{391}$

Decimal Form:

$122.76214833 \dots$