Calculus Examples

$lim_{x \to 2} \frac{1}{2} x^{5} - 2 x^{3} + 3 x^{2}$

Step 1

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

$lim_{x \to 2} \frac{1}{2} x^{5} - lim_{x \to 2} 2 x^{3} + lim_{x \to 2} 3 x^{2}$

Step 2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 2} x^{5} - lim_{x \to 2} 2 x^{3} + lim_{x \to 2} 3 x^{2}$

Step 3

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Cancel the common factor of $2$ .

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Step 9.1.1.1

Factor $2$ out of $2^{5}$ .

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

Step 9.1.1.2

Cancel the common factor.

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

Step 9.1.1.3

Rewrite the expression.

$2^{4} - 2 \cdot 2^{3} + 3 \cdot 2^{2}$

Step 9.1.2

Raise $2$ to the power of $4$ .

$16 - 2 \cdot 2^{3} + 3 \cdot 2^{2}$

Step 9.1.3

Raise $2$ to the power of $3$ .

$16 - 2 \cdot 8 + 3 \cdot 2^{2}$

Step 9.1.4

Multiply $- 2$ by $8$ .

$16 - 16 + 3 \cdot 2^{2}$

Step 9.1.5

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

$16 - 16 + 3 \cdot 4$

Step 9.1.6

Multiply $3$ by $4$ .

$16 - 16 + 12$

Step 9.2

Subtract $16$ from $16$ .

$0 + 12$

Step 9.3

Add $0$ and $12$ .

$12$