Calculus Examples

$lim_{x \to 0} {(5 x^{4} - 3 x^{3} + 7 x + 2)}^{5}$

Step 1

Move the exponent $5$ from ${(5 x^{4} - 3 x^{3} + 7 x + 2)}^{5}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 0} 5 x^{4} - 3 x^{3} + 7 x + 2)}^{5}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

${(lim_{x \to 0} 5 x^{4} - lim_{x \to 0} 3 x^{3} + lim_{x \to 0} 7 x + lim_{x \to 0} 2)}^{5}$

Step 3

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

${(5 lim_{x \to 0} x^{4} - lim_{x \to 0} 3 x^{3} + lim_{x \to 0} 7 x + lim_{x \to 0} 2)}^{5}$

Step 4

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

${(5 {(lim_{x \to 0} x)}^{4} - lim_{x \to 0} 3 x^{3} + lim_{x \to 0} 7 x + lim_{x \to 0} 2)}^{5}$

Step 5

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

${(5 {(lim_{x \to 0} x)}^{4} - 3 lim_{x \to 0} x^{3} + lim_{x \to 0} 7 x + lim_{x \to 0} 2)}^{5}$

Step 6

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

${(5 {(lim_{x \to 0} x)}^{4} - 3 {(lim_{x \to 0} x)}^{3} + lim_{x \to 0} 7 x + lim_{x \to 0} 2)}^{5}$

Step 7

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

${(5 {(lim_{x \to 0} x)}^{4} - 3 {(lim_{x \to 0} x)}^{3} + 7 lim_{x \to 0} x + lim_{x \to 0} 2)}^{5}$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

${(5 {(lim_{x \to 0} x)}^{4} - 3 {(lim_{x \to 0} x)}^{3} + 7 lim_{x \to 0} x + 2)}^{5}$

Step 9

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

${(5 \cdot 0^{4} - 3 {(lim_{x \to 0} x)}^{3} + 7 lim_{x \to 0} x + 2)}^{5}$

Step 9.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

${(5 \cdot 0^{4} - 3 \cdot 0^{3} + 7 lim_{x \to 0} x + 2)}^{5}$

Step 9.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

${(5 \cdot 0^{4} - 3 \cdot 0^{3} + 7 \cdot 0 + 2)}^{5}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Raising $0$ to any positive power yields $0$ .

${(5 \cdot 0 - 3 \cdot 0^{3} + 7 \cdot 0 + 2)}^{5}$

Step 10.1.2

Multiply $5$ by $0$ .

${(0 - 3 \cdot 0^{3} + 7 \cdot 0 + 2)}^{5}$

Step 10.1.3

Raising $0$ to any positive power yields $0$ .

${(0 - 3 \cdot 0 + 7 \cdot 0 + 2)}^{5}$

Step 10.1.4

Multiply $- 3$ by $0$ .

${(0 + 0 + 7 \cdot 0 + 2)}^{5}$

Step 10.1.5

Multiply $7$ by $0$ .

${(0 + 0 + 0 + 2)}^{5}$

Step 10.2

Add $0$ and $0$ .

${(0 + 0 + 2)}^{5}$

Step 10.3

Add $0$ and $0$ .

${(0 + 2)}^{5}$

Step 10.4

Add $0$ and $2$ .

$2^{5}$

Step 10.5

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

$32$