Calculus Examples

$f (x) = \frac{2}{e^{x} - e^{- x}}$

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

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

Solve for $x$ .

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Move $- e^{- x}$ to the right side of the equation by adding it to both sides.

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

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = - x$

Solve for $x$ .

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Move all terms containing $x$ to the left side of the equation.

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Add $x$ to both sides of the equation.

$x + x = 0$

Add $x$ and $x$ .

$2 x = 0$

Divide each term in $2 x = 0$ by $2$ and simplify.

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Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Simplify the right side.

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Divide $0$ by $2$ .

$x = 0$

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Since the domain is not all real numbers, $f (x) = \frac{2}{e^{x} - e^{- x}}$ is not continuous over all real numbers.

Not continuous