Calculus Examples

$lim_{x \to e} \frac{x + e}{x - e x}$

Step 1

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

$\frac{lim_{x \to e} x + e}{lim_{x \to e} x - e x}$

Step 2

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

$\frac{lim_{x \to e} x + lim_{x \to e} e}{lim_{x \to e} x - e x}$

Step 3

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

$\frac{lim_{x \to e} x + e}{lim_{x \to e} x - e x}$

Step 4

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

$\frac{lim_{x \to e} x + e}{lim_{x \to e} x - lim_{x \to e} e x}$

Step 5

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

$\frac{lim_{x \to e} x + e}{lim_{x \to e} x - e lim_{x \to e} x}$

Step 6

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

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Step 6.1

Evaluate the limit of $x$ by plugging in $e$ for $x$ .

$\frac{e + e}{lim_{x \to e} x - e lim_{x \to e} x}$

Step 6.2

Evaluate the limit of $x$ by plugging in $e$ for $x$ .

$\frac{e + e}{e - e lim_{x \to e} x}$

Step 6.3

Evaluate the limit of $x$ by plugging in $e$ for $x$ .

$\frac{e + e}{e - e e}$

Step 7

Simplify the answer.

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Step 7.1

Add $e$ and $e$ .

$\frac{2 e}{e - e e}$

Step 7.2

Multiply $- e e$ .

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Step 7.2.1

Raise $e$ to the power of $1$ .

$\frac{2 e}{e - (e^{1} e)}$

Step 7.2.2

Raise $e$ to the power of $1$ .

$\frac{2 e}{e - (e^{1} e^{1})}$

Step 7.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.2.4

Add $1$ and $1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.16395341 \dots$