Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $2$ by $2^{4}$ by adding the exponents.

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Step 9.1.1.1

Multiply $2$ by $2^{4}$ .

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Step 9.1.1.1.1

Raise $2$ to the power of $1$ .

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

Step 9.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9.1.1.2

Add $1$ and $4$ .

$2^{5} - 3 \cdot 2^{3} + 4 \cdot 2 - 1 \cdot 1$

Step 9.1.2

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

$32 - 3 \cdot 2^{3} + 4 \cdot 2 - 1 \cdot 1$

Step 9.1.3

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

$32 - 3 \cdot 8 + 4 \cdot 2 - 1 \cdot 1$

Step 9.1.4

Multiply $- 3$ by $8$ .

$32 - 24 + 4 \cdot 2 - 1 \cdot 1$

Step 9.1.5

Multiply $4$ by $2$ .

$32 - 24 + 8 - 1 \cdot 1$

Step 9.1.6

Multiply $- 1$ by $1$ .

$32 - 24 + 8 - 1$

Step 9.2

Subtract $24$ from $32$ .

$8 + 8 - 1$

Step 9.3

Add $8$ and $8$ .

$16 - 1$

Step 9.4

Subtract $1$ from $16$ .

$15$