Calculus Examples

$lim_{x \to \frac{3}{2}} (4 x^{2}) + 6 x + 9$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{3}{2}$ .

$lim_{x \to \frac{3}{2}} 4 x^{2} + lim_{x \to \frac{3}{2}} 6 x + lim_{x \to \frac{3}{2}} 9$

Step 2

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

$4 lim_{x \to \frac{3}{2}} x^{2} + lim_{x \to \frac{3}{2}} 6 x + lim_{x \to \frac{3}{2}} 9$

Step 3

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

$4 {(lim_{x \to \frac{3}{2}} x)}^{2} + lim_{x \to \frac{3}{2}} 6 x + lim_{x \to \frac{3}{2}} 9$

Step 4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$4 {(lim_{x \to \frac{3}{2}} x)}^{2} + 6 lim_{x \to \frac{3}{2}} x + lim_{x \to \frac{3}{2}} 9$

Step 5

Evaluate the limit of $9$ which is constant as $x$ approaches $\frac{3}{2}$ .

$4 {(lim_{x \to \frac{3}{2}} x)}^{2} + 6 lim_{x \to \frac{3}{2}} x + 9$

Step 6

Evaluate the limits by plugging in $\frac{3}{2}$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $\frac{3}{2}$ for $x$ .

$4 {(\frac{3}{2})}^{2} + 6 lim_{x \to \frac{3}{2}} x + 9$

Step 6.2

Evaluate the limit of $x$ by plugging in $\frac{3}{2}$ for $x$ .

$4 {(\frac{3}{2})}^{2} + 6 (\frac{3}{2}) + 9$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the product rule to $\frac{3}{2}$ .

$4 \frac{3^{2}}{2^{2}} + 6 (\frac{3}{2}) + 9$

Step 7.1.2

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

$4 \frac{9}{2^{2}} + 6 (\frac{3}{2}) + 9$

Step 7.1.3

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

$4 (\frac{9}{4}) + 6 (\frac{3}{2}) + 9$

Step 7.1.4

Cancel the common factor of $4$ .

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Step 7.1.4.1

Cancel the common factor.

$4 (\frac{9}{4}) + 6 (\frac{3}{2}) + 9$

Step 7.1.4.2

Rewrite the expression.

$9 + 6 (\frac{3}{2}) + 9$

Step 7.1.5

Cancel the common factor of $2$ .

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Step 7.1.5.1

Factor $2$ out of $6$ .

$9 + 2 (3) \frac{3}{2} + 9$

Step 7.1.5.2

Cancel the common factor.

$9 + 2 \cdot 3 \frac{3}{2} + 9$

Step 7.1.5.3

Rewrite the expression.

$9 + 3 \cdot 3 + 9$

Step 7.1.6

Multiply $3$ by $3$ .

$9 + 9 + 9$

Step 7.2

Add $9$ and $9$ .

$18 + 9$

Step 7.3

Add $18$ and $9$ .

$27$