Calculus Examples

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

Step 1

Move the exponent $3$ from ${(x^{3} + x^{2} - 5 x - 3)}^{3}$ outside the limit using the Limits Power Rule.

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

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

Step 7

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 7.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 7.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Raise $2$ to the power of $3$ .

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

Step 8.1.2

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

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

Step 8.1.3

Multiply $- 5$ by $2$ .

${(8 + 4 - 10 - 1 \cdot 3)}^{3}$

Step 8.1.4

Multiply $- 1$ by $3$ .

${(8 + 4 - 10 - 3)}^{3}$

Step 8.2

Add $8$ and $4$ .

${(12 - 10 - 3)}^{3}$

Step 8.3

Subtract $10$ from $12$ .

${(2 - 3)}^{3}$

Step 8.4

Subtract $3$ from $2$ .

${(- 1)}^{3}$

Step 8.5

Raise $- 1$ to the power of $3$ .

$- 1$