Calculus Examples

$lim_{x \to - 1} \frac{3 x - 9 x^{2}}{2 \ln (- 1 - 2 x) - 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 1$ .

$\frac{lim_{x \to - 1} 3 x - 9 x^{2}}{lim_{x \to - 1} 2 \ln (- 1 - 2 x) - 2}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{lim_{x \to - 1} 3 x - lim_{x \to - 1} 9 x^{2}}{lim_{x \to - 1} 2 \ln (- 1 - 2 x) - 2}$

Step 3

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

$\frac{3 lim_{x \to - 1} x - lim_{x \to - 1} 9 x^{2}}{lim_{x \to - 1} 2 \ln (- 1 - 2 x) - 2}$

Step 4

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to - 1} x - 9 lim_{x \to - 1} x^{2}}{lim_{x \to - 1} 2 \ln (- 1 - 2 x) - 2}$

Step 5

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

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{lim_{x \to - 1} 2 \ln (- 1 - 2 x) - 2}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{lim_{x \to - 1} 2 \ln (- 1 - 2 x) - lim_{x \to - 1} 2}$

Step 7

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

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{2 lim_{x \to - 1} \ln (- 1 - 2 x) - lim_{x \to - 1} 2}$

Step 8

Move the limit inside the logarithm.

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{2 \ln (lim_{x \to - 1} - 1 - 2 x) - lim_{x \to - 1} 2}$

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{2 \ln (- lim_{x \to - 1} 1 - lim_{x \to - 1} 2 x) - lim_{x \to - 1} 2}$

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $- 1$ .

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{2 \ln (- 1 \cdot 1 - lim_{x \to - 1} 2 x) - lim_{x \to - 1} 2}$

Step 11

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

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{2 \ln (- 1 - 2 lim_{x \to - 1} x) - lim_{x \to - 1} 2}$

Step 12

Evaluate the limit of $2$ which is constant as $x$ approaches $- 1$ .

$\frac{3 lim_{x \to - 1} x - 9 {(lim_{x \to - 1} x)}^{2}}{2 \ln (- 1 - 2 lim_{x \to - 1} x) - 1 \cdot 2}$

Step 13

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

Tap for more steps...

Step 13.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{3 \cdot - 1 - 9 {(lim_{x \to - 1} x)}^{2}}{2 \ln (- 1 - 2 lim_{x \to - 1} x) - 1 \cdot 2}$

Step 13.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{3 \cdot - 1 - 9 {(- 1)}^{2}}{2 \ln (- 1 - 2 lim_{x \to - 1} x) - 1 \cdot 2}$

Step 13.3

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{3 \cdot - 1 - 9 {(- 1)}^{2}}{2 \ln (- 1 - 2 \cdot - 1) - 1 \cdot 2}$

Step 14

Simplify the answer.

Tap for more steps...

Step 14.1

Simplify the numerator.

Tap for more steps...

Step 14.1.1

Multiply $3$ by $- 1$ .

$\frac{- 3 - 9 {(- 1)}^{2}}{2 \ln (- 1 - 2 \cdot - 1) - 1 \cdot 2}$

Step 14.1.2

Raise $- 1$ to the power of $2$ .

$\frac{- 3 - 9 \cdot 1}{2 \ln (- 1 - 2 \cdot - 1) - 1 \cdot 2}$

Step 14.1.3

Multiply $- 9$ by $1$ .

$\frac{- 3 - 9}{2 \ln (- 1 - 2 \cdot - 1) - 1 \cdot 2}$

Step 14.1.4

Subtract $9$ from $- 3$ .

$\frac{- 12}{2 \ln (- 1 - 2 \cdot - 1) - 1 \cdot 2}$

Step 14.2

Simplify the denominator.

Tap for more steps...

Step 14.2.1

Multiply $- 2$ by $- 1$ .

$\frac{- 12}{2 \ln (- 1 + 2) - 1 \cdot 2}$

Step 14.2.2

Add $- 1$ and $2$ .

$\frac{- 12}{2 \ln (1) - 1 \cdot 2}$

Step 14.2.3

The natural logarithm of $1$ is $0$ .

$\frac{- 12}{2 \cdot 0 - 1 \cdot 2}$

Step 14.2.4

Multiply $2$ by $0$ .

$\frac{- 12}{0 - 1 \cdot 2}$

Step 14.2.5

Multiply $- 1$ by $2$ .

$\frac{- 12}{0 - 2}$

Step 14.2.6

Subtract $2$ from $0$ .

$\frac{- 12}{- 2}$

Step 14.3

Divide $- 12$ by $- 2$ .

$6$