Calculus Examples

$lim_{x \to 1} x^{3} + 3 x^{2} - 2 x - 17$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$lim_{x \to 1} x^{3} + lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 17$

Step 2

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

${(lim_{x \to 1} x)}^{3} + lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 17$

Step 3

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

${(lim_{x \to 1} x)}^{3} + 3 lim_{x \to 1} x^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 17$

Step 4

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

${(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 2 x - lim_{x \to 1} 17$

Step 5

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

${(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - lim_{x \to 1} 17$

Step 6

Evaluate the limit of $17$ which is constant as $x$ approaches $1$ .

${(lim_{x \to 1} x)}^{3} + 3 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 17$

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$1^{3} + 3 {(lim_{x \to 1} x)}^{2} - 2 lim_{x \to 1} x - 1 \cdot 17$

Step 7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$1^{3} + 3 \cdot 1^{2} - 2 lim_{x \to 1} x - 1 \cdot 17$

Step 7.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$1^{3} + 3 \cdot 1^{2} - 2 \cdot 1 - 1 \cdot 17$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

One to any power is one.

$1 + 3 \cdot 1^{2} - 2 \cdot 1 - 1 \cdot 17$

Step 8.1.2

One to any power is one.

$1 + 3 \cdot 1 - 2 \cdot 1 - 1 \cdot 17$

Step 8.1.3

Multiply $3$ by $1$ .

$1 + 3 - 2 \cdot 1 - 1 \cdot 17$

Step 8.1.4

Multiply $- 2$ by $1$ .

$1 + 3 - 2 - 1 \cdot 17$

Step 8.1.5

Multiply $- 1$ by $17$ .

$1 + 3 - 2 - 17$

Step 8.2

Add $1$ and $3$ .

$4 - 2 - 17$

Step 8.3

Subtract $2$ from $4$ .

$2 - 17$

Step 8.4

Subtract $17$ from $2$ .

$- 15$