Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Move the limit into the exponent.

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

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$5 {(lim_{x \to 0} x)}^{3} - 7 {(lim_{x \to 0} x)}^{2} + 2^{lim_{x \to 0} x} - 1 \cdot 2$

Step 8

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

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Step 8.1

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

$5 \cdot 0^{3} - 7 {(lim_{x \to 0} x)}^{2} + 2^{lim_{x \to 0} x} - 1 \cdot 2$

Step 8.2

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

$5 \cdot 0^{3} - 7 \cdot 0^{2} + 2^{lim_{x \to 0} x} - 1 \cdot 2$

Step 8.3

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

$5 \cdot 0^{3} - 7 \cdot 0^{2} + 2^{0} - 1 \cdot 2$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$5 \cdot 0 - 7 \cdot 0^{2} + 2^{0} - 1 \cdot 2$

Step 9.1.2

Multiply $5$ by $0$ .

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

Step 9.1.3

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

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

Step 9.1.4

Multiply $- 7$ by $0$ .

$0 + 0 + 2^{0} - 1 \cdot 2$

Step 9.1.5

Anything raised to $0$ is $1$ .

$0 + 0 + 1 - 1 \cdot 2$

Step 9.1.6

Multiply $- 1$ by $2$ .

$0 + 0 + 1 - 2$

Step 9.2

Add $0$ and $0$ .

$0 + 1 - 2$

Step 9.3

Add $0$ and $1$ .

$1 - 2$

Step 9.4

Subtract $2$ from $1$ .

$- 1$