Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

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

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 $2$ .

$3 \cdot 4 - 5 \cdot 2 + 2$

Step 7.1.2

Multiply $3$ by $4$ .

$12 - 5 \cdot 2 + 2$

Step 7.1.3

Multiply $- 5$ by $2$ .

$12 - 10 + 2$

Step 7.2

Subtract $10$ from $12$ .

$2 + 2$

Step 7.3

Add $2$ and $2$ .

$4$