Calculus Examples

$lim_{x \to 8} \frac{5 - 2 x^{\frac{3}{2}}}{3 x^{2} - 4}$

Step 1

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

$\frac{lim_{x \to 8} 5 - 2 x^{\frac{3}{2}}}{lim_{x \to 8} 3 x^{2} - 4}$

Step 2

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

$\frac{lim_{x \to 8} 5 - lim_{x \to 8} 2 x^{\frac{3}{2}}}{lim_{x \to 8} 3 x^{2} - 4}$

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{5 - lim_{x \to 8} 2 x^{\frac{3}{2}}}{lim_{x \to 8} 3 x^{2} - 4}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 - 2 lim_{x \to 8} x^{\frac{3}{2}}}{lim_{x \to 8} 3 x^{2} - 4}$

Step 5

Move the exponent $\frac{3}{2}$ from $x^{\frac{3}{2}}$ outside the limit using the Limits Power Rule.

$\frac{5 - 2 {(lim_{x \to 8} x)}^{\frac{3}{2}}}{lim_{x \to 8} 3 x^{2} - 4}$

Step 6

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

$\frac{5 - 2 {(lim_{x \to 8} x)}^{\frac{3}{2}}}{lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 4}$

Step 7

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

$\frac{5 - 2 {(lim_{x \to 8} x)}^{\frac{3}{2}}}{3 lim_{x \to 8} x^{2} - lim_{x \to 8} 4}$

Step 8

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

$\frac{5 - 2 {(lim_{x \to 8} x)}^{\frac{3}{2}}}{3 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 4}$

Step 9

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{5 - 2 {(lim_{x \to 8} x)}^{\frac{3}{2}}}{3 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}$

Step 10

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

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Step 10.1

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

$\frac{5 - 2 \cdot 8^{\frac{3}{2}}}{3 {(lim_{x \to 8} x)}^{2} - 1 \cdot 4}$

Step 10.2

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

$\frac{5 - 2 \cdot 8^{\frac{3}{2}}}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11

Simplify the answer.

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Step 11.1

Multiply $- 2 \cdot 8^{\frac{3}{2}}$ .

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Step 11.1.1

Factor out negative.

$\frac{5 - (2 \cdot 8^{\frac{3}{2}})}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.2

Rewrite $8$ as $2^{3}$ .

$\frac{5 - (2 \cdot {(2^{3})}^{\frac{3}{2}})}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.3

Multiply the exponents in ${(2^{3})}^{\frac{3}{2}}$ .

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Step 11.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5 - (2 \cdot 2^{3 (\frac{3}{2})})}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.3.2

Multiply $3 (\frac{3}{2})$ .

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Step 11.1.3.2.1

Combine $3$ and $\frac{3}{2}$ .

$\frac{5 - (2 \cdot 2^{\frac{3 \cdot 3}{2}})}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.3.2.2

Multiply $3$ by $3$ .

$\frac{5 - (2 \cdot 2^{\frac{9}{2}})}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 - 2^{1 + \frac{9}{2}}}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.5

Write $1$ as a fraction with a common denominator.

$\frac{5 - 2^{\frac{2}{2} + \frac{9}{2}}}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.6

Combine the numerators over the common denominator.

$\frac{5 - 2^{\frac{2 + 9}{2}}}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.1.7

Add $2$ and $9$ .

$\frac{5 - 2^{\frac{11}{2}}}{3 \cdot 8^{2} - 1 \cdot 4}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

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

$\frac{5 - 2^{\frac{11}{2}}}{3 \cdot 64 - 1 \cdot 4}$

Step 11.2.2

Multiply $3$ by $64$ .

$\frac{5 - 2^{\frac{11}{2}}}{192 - 1 \cdot 4}$

Step 11.2.3

Multiply $- 1$ by $4$ .

$\frac{5 - 2^{\frac{11}{2}}}{192 - 4}$

Step 11.2.4

Subtract $4$ from $192$ .

$\frac{5 - 2^{\frac{11}{2}}}{188}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{5 - 2^{\frac{11}{2}}}{188}$

Decimal Form:

$- 0.21412145 \dots$