Calculus Examples

$lim_{x \to 2} x^{3} + 5 x^{2} - 7 x + 1$

Step 1

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

$lim_{x \to 2} x^{3} + lim_{x \to 2} 5 x^{2} - lim_{x \to 2} 7 x + lim_{x \to 2} 1$

Step 2

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

${(lim_{x \to 2} x)}^{3} + lim_{x \to 2} 5 x^{2} - lim_{x \to 2} 7 x + lim_{x \to 2} 1$

Step 3

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

${(lim_{x \to 2} x)}^{3} + 5 lim_{x \to 2} x^{2} - lim_{x \to 2} 7 x + lim_{x \to 2} 1$

Step 4

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

${(lim_{x \to 2} x)}^{3} + 5 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 7 x + lim_{x \to 2} 1$

Step 5

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

${(lim_{x \to 2} x)}^{3} + 5 {(lim_{x \to 2} x)}^{2} - 7 lim_{x \to 2} x + lim_{x \to 2} 1$

Step 6

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

${(lim_{x \to 2} x)}^{3} + 5 {(lim_{x \to 2} x)}^{2} - 7 lim_{x \to 2} x + 1$

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 $x$ .

$2^{3} + 5 {(lim_{x \to 2} x)}^{2} - 7 lim_{x \to 2} x + 1$

Step 7.2

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

$2^{3} + 5 \cdot 2^{2} - 7 lim_{x \to 2} x + 1$

Step 7.3

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

$2^{3} + 5 \cdot 2^{2} - 7 \cdot 2 + 1$

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 $3$ .

$8 + 5 \cdot 2^{2} - 7 \cdot 2 + 1$

Step 8.1.2

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

$8 + 5 \cdot 4 - 7 \cdot 2 + 1$

Step 8.1.3

Multiply $5$ by $4$ .

$8 + 20 - 7 \cdot 2 + 1$

Step 8.1.4

Multiply $- 7$ by $2$ .

$8 + 20 - 14 + 1$

Step 8.2

Add $8$ and $20$ .

$28 - 14 + 1$

Step 8.3

Subtract $14$ from $28$ .

$14 + 1$

Step 8.4

Add $14$ and $1$ .

$15$