Calculus Examples

$lim_{x \to 0} 5 + 7 x - 3 x^{2}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} 5 + lim_{x \to 0} 7 x - lim_{x \to 0} 3 x^{2}$

Step 2

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$5 + lim_{x \to 0} 7 x - lim_{x \to 0} 3 x^{2}$

Step 3

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

$5 + 7 lim_{x \to 0} x - lim_{x \to 0} 3 x^{2}$

Step 4

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

$5 + 7 lim_{x \to 0} x - 3 lim_{x \to 0} x^{2}$

Step 5

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

$5 + 7 lim_{x \to 0} x - 3 {(lim_{x \to 0} x)}^{2}$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$5 + 7 \cdot 0 - 3 {(lim_{x \to 0} x)}^{2}$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$5 + 7 \cdot 0 - 3 \cdot 0^{2}$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $7$ by $0$ .

$5 + 0 - 3 \cdot 0^{2}$

Step 7.1.2

Raising $0$ to any positive power yields $0$ .

$5 + 0 - 3 \cdot 0$

Step 7.1.3

Multiply $- 3$ by $0$ .

$5 + 0 + 0$

Step 7.2

Add $5$ and $0$ .

$5 + 0$

Step 7.3

Add $5$ and $0$ .

$5$