Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$2^{3} - 2^{2} - 1 \cdot 4$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$8 - 2^{2} - 1 \cdot 4$

Step 6.1.2

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

$8 - 1 \cdot 4 - 1 \cdot 4$

Step 6.1.3

Multiply $- 1$ by $4$ .

$8 - 4 - 1 \cdot 4$

Step 6.1.4

Multiply $- 1$ by $4$ .

$8 - 4 - 4$

Step 6.2

Subtract $4$ from $8$ .

$4 - 4$

Step 6.3

Subtract $4$ from $4$ .

$0$