Calculus Examples

$lim_{x \to \frac{π}{2}} x^{3} + 5 x^{2} - 2 x - 15$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$lim_{x \to \frac{π}{2}} x^{3} + lim_{x \to \frac{π}{2}} 5 x^{2} - lim_{x \to \frac{π}{2}} 2 x - lim_{x \to \frac{π}{2}} 15$

Step 2

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

${(lim_{x \to \frac{π}{2}} x)}^{3} + lim_{x \to \frac{π}{2}} 5 x^{2} - lim_{x \to \frac{π}{2}} 2 x - lim_{x \to \frac{π}{2}} 15$

Step 3

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

${(lim_{x \to \frac{π}{2}} x)}^{3} + 5 lim_{x \to \frac{π}{2}} x^{2} - lim_{x \to \frac{π}{2}} 2 x - lim_{x \to \frac{π}{2}} 15$

Step 4

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

${(lim_{x \to \frac{π}{2}} x)}^{3} + 5 {(lim_{x \to \frac{π}{2}} x)}^{2} - lim_{x \to \frac{π}{2}} 2 x - lim_{x \to \frac{π}{2}} 15$

Step 5

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

${(lim_{x \to \frac{π}{2}} x)}^{3} + 5 {(lim_{x \to \frac{π}{2}} x)}^{2} - 2 lim_{x \to \frac{π}{2}} x - lim_{x \to \frac{π}{2}} 15$

Step 6

Evaluate the limit of $15$ which is constant as $x$ approaches $\frac{π}{2}$ .

${(lim_{x \to \frac{π}{2}} x)}^{3} + 5 {(lim_{x \to \frac{π}{2}} x)}^{2} - 2 lim_{x \to \frac{π}{2}} x - 1 \cdot 15$

Step 7

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

${(\frac{π}{2})}^{3} + 5 {(lim_{x \to \frac{π}{2}} x)}^{2} - 2 lim_{x \to \frac{π}{2}} x - 1 \cdot 15$

Step 7.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

${(\frac{π}{2})}^{3} + 5 {(\frac{π}{2})}^{2} - 2 lim_{x \to \frac{π}{2}} x - 1 \cdot 15$

Step 7.3

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

${(\frac{π}{2})}^{3} + 5 {(\frac{π}{2})}^{2} - 2 \frac{π}{2} - 1 \cdot 15$

Step 8

Simplify each term.

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Step 8.1

Apply the product rule to $\frac{π}{2}$ .

$\frac{π^{3}}{2^{3}} + 5 {(\frac{π}{2})}^{2} - 2 \frac{π}{2} - 1 \cdot 15$

Step 8.2

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

$\frac{π^{3}}{8} + 5 {(\frac{π}{2})}^{2} - 2 \frac{π}{2} - 1 \cdot 15$

Step 8.3

Apply the product rule to $\frac{π}{2}$ .

$\frac{π^{3}}{8} + 5 \frac{π^{2}}{2^{2}} - 2 \frac{π}{2} - 1 \cdot 15$

Step 8.4

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

$\frac{π^{3}}{8} + 5 \frac{π^{2}}{4} - 2 \frac{π}{2} - 1 \cdot 15$

Step 8.5

Combine $5$ and $\frac{π^{2}}{4}$ .

$\frac{π^{3}}{8} + \frac{5 π^{2}}{4} - 2 \frac{π}{2} - 1 \cdot 15$

Step 8.6

Cancel the common factor of $2$ .

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Step 8.6.1

Factor $2$ out of $- 2$ .

$\frac{π^{3}}{8} + \frac{5 π^{2}}{4} + 2 (- 1) \frac{π}{2} - 1 \cdot 15$

Step 8.6.2

Cancel the common factor.

$\frac{π^{3}}{8} + \frac{5 π^{2}}{4} + 2 \cdot - 1 \frac{π}{2} - 1 \cdot 15$

Step 8.6.3

Rewrite the expression.

$\frac{π^{3}}{8} + \frac{5 π^{2}}{4} - π - 1 \cdot 15$

Step 8.7

Multiply $- 1$ by $15$ .

$\frac{π^{3}}{8} + \frac{5 π^{2}}{4} - π - 15$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{π^{3}}{8} + \frac{5 π^{2}}{4} - π - 15$

Decimal Form:

$- 1.92880256 \dots$