Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 8

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 8.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 8.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 8.4

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$81 + 3^{3} - 4 \cdot 3^{2} - 2 \cdot 3 + 1$

Step 9.1.2

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

$81 + 27 - 4 \cdot 3^{2} - 2 \cdot 3 + 1$

Step 9.1.3

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

$81 + 27 - 4 \cdot 9 - 2 \cdot 3 + 1$

Step 9.1.4

Multiply $- 4$ by $9$ .

$81 + 27 - 36 - 2 \cdot 3 + 1$

Step 9.1.5

Multiply $- 2$ by $3$ .

$81 + 27 - 36 - 6 + 1$

Step 9.2

Add $81$ and $27$ .

$108 - 36 - 6 + 1$

Step 9.3

Subtract $36$ from $108$ .

$72 - 6 + 1$

Step 9.4

Subtract $6$ from $72$ .

$66 + 1$

Step 9.5

Add $66$ and $1$ .

$67$