Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

$8 - 2 \cdot 2^{2} - 1 \cdot 3$

Step 7.1.2

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

$8 - 2 \cdot 4 - 1 \cdot 3$

Step 7.1.3

Multiply $- 2$ by $4$ .

$8 - 8 - 1 \cdot 3$

Step 7.1.4

Multiply $- 1$ by $3$ .

$8 - 8 - 3$

Step 7.2

Subtract $8$ from $8$ .

$0 - 3$

Step 7.3

Subtract $3$ from $0$ .

$- 3$