Calculus Examples

$\frac{1 - e^{x^{2}}}{1 - e^{1 - x^{2}}}$

Step 1

Set the denominator in $\frac{1 - e^{x^{2}}}{1 - e^{1 - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$1 - e^{1 - x^{2}} = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $1$ from both sides of the equation.

$- e^{1 - x^{2}} = - 1$

Step 2.2

Divide each term in $- e^{1 - x^{2}} = - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- e^{1 - x^{2}} = - 1$ by $- 1$ .

$\frac{- e^{1 - x^{2}}}{- 1} = \frac{- 1}{- 1}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{e^{1 - x^{2}}}{1} = \frac{- 1}{- 1}$

Step 2.2.2.2

Divide $e^{1 - x^{2}}$ by $1$ .

$e^{1 - x^{2}} = \frac{- 1}{- 1}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$e^{1 - x^{2}} = 1$

Step 2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{1 - x^{2}}) = \ln (1)$

Step 2.4

Expand the left side.

Tap for more steps...

Step 2.4.1

Expand $\ln (e^{1 - x^{2}})$ by moving $1 - x^{2}$ outside the logarithm.

$(1 - x^{2}) \ln (e) = \ln (1)$

Step 2.4.2

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

$(1 - x^{2}) \cdot 1 = \ln (1)$

Step 2.4.3

Multiply $1 - x^{2}$ by $1$ .

$1 - x^{2} = \ln (1)$

Step 2.5

Simplify the right side.

Tap for more steps...

Step 2.5.1

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

$1 - x^{2} = 0$

Step 2.6

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 2.7

Divide each term in $- x^{2} = - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 2.7.1

Divide each term in $- x^{2} = - 1$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 1}{- 1}$

Step 2.7.2

Simplify the left side.

Tap for more steps...

Step 2.7.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 1}{- 1}$

Step 2.7.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{- 1}$

Step 2.7.3

Simplify the right side.

Tap for more steps...

Step 2.7.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 2.8

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 2.9

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.10

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.10.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 2.10.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 2.10.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 1}$

Step 4