Calculus Examples

$lim_{x \to \frac{1}{2}} \frac{2 x + 1}{6 x^{2} + 3 x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} 2 x + 1}{lim_{x \to \frac{1}{2}} 6 x^{2} + 3 x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} 2 x + lim_{x \to \frac{1}{2}} 1}{lim_{x \to \frac{1}{2}} 6 x^{2} + 3 x}$

Step 3

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

$\frac{2 lim_{x \to \frac{1}{2}} x + lim_{x \to \frac{1}{2}} 1}{lim_{x \to \frac{1}{2}} 6 x^{2} + 3 x}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{2 lim_{x \to \frac{1}{2}} x + 1}{lim_{x \to \frac{1}{2}} 6 x^{2} + 3 x}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{2 lim_{x \to \frac{1}{2}} x + 1}{lim_{x \to \frac{1}{2}} 6 x^{2} + lim_{x \to \frac{1}{2}} 3 x}$

Step 6

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

$\frac{2 lim_{x \to \frac{1}{2}} x + 1}{6 lim_{x \to \frac{1}{2}} x^{2} + lim_{x \to \frac{1}{2}} 3 x}$

Step 7

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

$\frac{2 lim_{x \to \frac{1}{2}} x + 1}{6 {(lim_{x \to \frac{1}{2}} x)}^{2} + lim_{x \to \frac{1}{2}} 3 x}$

Step 8

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

$\frac{2 lim_{x \to \frac{1}{2}} x + 1}{6 {(lim_{x \to \frac{1}{2}} x)}^{2} + 3 lim_{x \to \frac{1}{2}} x}$

Step 9

Evaluate the limits by plugging in $\frac{1}{2}$ for all occurrences of $x$ .

Tap for more steps...

Step 9.1

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{2 (\frac{1}{2}) + 1}{6 {(lim_{x \to \frac{1}{2}} x)}^{2} + 3 lim_{x \to \frac{1}{2}} x}$

Step 9.2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{2 (\frac{1}{2}) + 1}{6 {(\frac{1}{2})}^{2} + 3 lim_{x \to \frac{1}{2}} x}$

Step 9.3

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{2 (\frac{1}{2}) + 1}{6 {(\frac{1}{2})}^{2} + 3 (\frac{1}{2})}$

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.1.1.1

Cancel the common factor.

$\frac{2 (\frac{1}{2}) + 1}{6 {(\frac{1}{2})}^{2} + 3 (\frac{1}{2})}$

Step 10.1.1.2

Rewrite the expression.

$\frac{1 + 1}{6 {(\frac{1}{2})}^{2} + 3 (\frac{1}{2})}$

Step 10.1.2

Add $1$ and $1$ .

$\frac{2}{6 {(\frac{1}{2})}^{2} + 3 (\frac{1}{2})}$

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{2}{6 \frac{1^{2}}{2^{2}} + 3 (\frac{1}{2})}$

Step 10.2.2

One to any power is one.

$\frac{2}{6 \frac{1}{2^{2}} + 3 (\frac{1}{2})}$

Step 10.2.3

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

$\frac{2}{6 (\frac{1}{4}) + 3 (\frac{1}{2})}$

Step 10.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.4.1

Factor $2$ out of $6$ .

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

Step 10.2.4.2

Factor $2$ out of $4$ .

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

Step 10.2.4.3

Cancel the common factor.

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

Step 10.2.4.4

Rewrite the expression.

$\frac{2}{3 (\frac{1}{2}) + 3 (\frac{1}{2})}$

Step 10.2.5

Combine $3$ and $\frac{1}{2}$ .

$\frac{2}{\frac{3}{2} + 3 (\frac{1}{2})}$

Step 10.2.6

Combine $3$ and $\frac{1}{2}$ .

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

Step 10.2.7

Combine the numerators over the common denominator.

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

Step 10.2.8

Add $3$ and $3$ .

$\frac{2}{\frac{6}{2}}$

Step 10.2.9

Divide $6$ by $2$ .

$\frac{2}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$