Calculus Examples

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

Step 1

Move the exponent $4$ from ${(2 x^{2} + 3 x + 1)}^{4}$ outside the limit using the Limits Power Rule.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $- 2$ .

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

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 $x$ .

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

Step 7.2

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

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

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 $2$ .

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

Step 8.1.2

Multiply $2$ by $4$ .

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

Step 8.1.3

Multiply $3$ by $- 2$ .

${(8 - 6 + 1)}^{4}$

Step 8.2

Subtract $6$ from $8$ .

${(2 + 1)}^{4}$

Step 8.3

Add $2$ and $1$ .

$3^{4}$

Step 8.4

Raise $3$ to the power of $4$ .

$81$