Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $- 2$ by $4$ .

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

Step 7.1.3

Multiply $3$ by $2$ .

$- 8 + 6 - 1 \cdot 1$

Step 7.1.4

Multiply $- 1$ by $1$ .

$- 8 + 6 - 1$

Step 7.2

Add $- 8$ and $6$ .

$- 2 - 1$

Step 7.3

Subtract $1$ from $- 2$ .

$- 3$