Calculus Examples

$lim_{x \to 0} (4 x^{2} - 3) (7 x^{3} + 2 x)$

Step 1

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

$lim_{x \to 0} 4 x^{2} - 3 \cdot lim_{x \to 0} 7 x^{3} + 2 x$

Step 2

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

$(lim_{x \to 0} 4 x^{2} - lim_{x \to 0} 3) \cdot lim_{x \to 0} 7 x^{3} + 2 x$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$(4 lim_{x \to 0} x^{2} - lim_{x \to 0} 3) \cdot lim_{x \to 0} 7 x^{3} + 2 x$

Step 4

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

$(4 {(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 3) \cdot lim_{x \to 0} 7 x^{3} + 2 x$

Step 5

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$(4 {(lim_{x \to 0} x)}^{2} - 1 \cdot 3) \cdot lim_{x \to 0} 7 x^{3} + 2 x$

Step 6

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

$(4 {(lim_{x \to 0} x)}^{2} - 1 \cdot 3) \cdot (lim_{x \to 0} 7 x^{3} + lim_{x \to 0} 2 x)$

Step 7

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

$(4 {(lim_{x \to 0} x)}^{2} - 1 \cdot 3) \cdot (7 lim_{x \to 0} x^{3} + lim_{x \to 0} 2 x)$

Step 8

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

$(4 {(lim_{x \to 0} x)}^{2} - 1 \cdot 3) \cdot (7 {(lim_{x \to 0} x)}^{3} + lim_{x \to 0} 2 x)$

Step 9

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

$(4 {(lim_{x \to 0} x)}^{2} - 1 \cdot 3) \cdot (7 {(lim_{x \to 0} x)}^{3} + 2 lim_{x \to 0} x)$

Step 10

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

Tap for more steps...

Step 10.1

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

$(4 \cdot 0^{2} - 1 \cdot 3) \cdot (7 {(lim_{x \to 0} x)}^{3} + 2 lim_{x \to 0} x)$

Step 10.2

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

$(4 \cdot 0^{2} - 1 \cdot 3) \cdot (7 \cdot 0^{3} + 2 lim_{x \to 0} x)$

Step 10.3

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

$(4 \cdot 0^{2} - 1 \cdot 3) \cdot (7 \cdot 0^{3} + 2 \cdot 0)$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

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

$(4 \cdot 0 - 1 \cdot 3) \cdot (7 \cdot 0^{3} + 2 \cdot 0)$

Step 11.1.2

Multiply $4$ by $0$ .

$(0 - 1 \cdot 3) \cdot (7 \cdot 0^{3} + 2 \cdot 0)$

Step 11.1.3

Multiply $- 1$ by $3$ .

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

Step 11.2

Subtract $3$ from $0$ .

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

Step 11.3

Simplify each term.

Tap for more steps...

Step 11.3.1

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

$- 3 \cdot (7 \cdot 0 + 2 \cdot 0)$

Step 11.3.2

Multiply $7$ by $0$ .

$- 3 \cdot (0 + 2 \cdot 0)$

Step 11.3.3

Multiply $2$ by $0$ .

$- 3 \cdot (0 + 0)$

Step 11.4

Add $0$ and $0$ .

$- 3 \cdot 0$

Step 11.5

Multiply $- 3$ by $0$ .

$0$