Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

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

Step 6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Evaluate the limit of $5$ which is constant as $x$ approaches $- 2$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

$3 {(- 2)}^{5} - 3 {(- 2)}^{4} + 4 {(- 2)}^{3} + {(- 2)}^{2} + 5$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Raise $- 2$ to the power of $5$ .

$3 \cdot - 32 - 3 {(- 2)}^{4} + 4 {(- 2)}^{3} + {(- 2)}^{2} + 5$

Step 11.1.2

Multiply $3$ by $- 32$ .

$- 96 - 3 {(- 2)}^{4} + 4 {(- 2)}^{3} + {(- 2)}^{2} + 5$

Step 11.1.3

Raise $- 2$ to the power of $4$ .

$- 96 - 3 \cdot 16 + 4 {(- 2)}^{3} + {(- 2)}^{2} + 5$

Step 11.1.4

Multiply $- 3$ by $16$ .

$- 96 - 48 + 4 {(- 2)}^{3} + {(- 2)}^{2} + 5$

Step 11.1.5

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

$- 96 - 48 + 4 \cdot - 8 + {(- 2)}^{2} + 5$

Step 11.1.6

Multiply $4$ by $- 8$ .

$- 96 - 48 - 32 + {(- 2)}^{2} + 5$

Step 11.1.7

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

$- 96 - 48 - 32 + 4 + 5$

Step 11.2

Subtract $48$ from $- 96$ .

$- 144 - 32 + 4 + 5$

Step 11.3

Subtract $32$ from $- 144$ .

$- 176 + 4 + 5$

Step 11.4

Add $- 176$ and $4$ .

$- 172 + 5$

Step 11.5

Add $- 172$ and $5$ .

$- 167$