Calculus Examples

$lim_{x \to 1^{+}} \frac{x}{1 - e^{\frac{x + 1}{x + 3}}}$

Step 1

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

$\frac{lim_{x \to 1^{+}} x}{lim_{x \to 1^{+}} 1 - e^{\frac{x + 1}{x + 3}}}$

Step 2

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

$\frac{lim_{x \to 1^{+}} x}{lim_{x \to 1^{+}} 1 - lim_{x \to 1^{+}} e^{\frac{x + 1}{x + 3}}}$

Step 3

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

$\frac{lim_{x \to 1^{+}} x}{1 - lim_{x \to 1^{+}} e^{\frac{x + 1}{x + 3}}}$

Step 4

Move the limit into the exponent.

$\frac{lim_{x \to 1^{+}} x}{1 - e^{lim_{x \to 1^{+}} \frac{x + 1}{x + 3}}}$

Step 5

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

$\frac{lim_{x \to 1^{+}} x}{1 - e^{\frac{lim_{x \to 1^{+}} x + 1}{lim_{x \to 1^{+}} x + 3}}}$

Step 6

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

$\frac{lim_{x \to 1^{+}} x}{1 - e^{\frac{lim_{x \to 1^{+}} x + lim_{x \to 1^{+}} 1}{lim_{x \to 1^{+}} x + 3}}}$

Step 7

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

$\frac{lim_{x \to 1^{+}} x}{1 - e^{\frac{lim_{x \to 1^{+}} x + 1}{lim_{x \to 1^{+}} x + 3}}}$

Step 8

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

$\frac{lim_{x \to 1^{+}} x}{1 - e^{\frac{lim_{x \to 1^{+}} x + 1}{lim_{x \to 1^{+}} x + lim_{x \to 1^{+}} 3}}}$

Step 9

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\frac{lim_{x \to 1^{+}} x}{1 - e^{\frac{lim_{x \to 1^{+}} x + 1}{lim_{x \to 1^{+}} x + 3}}}$

Step 10

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

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Step 10.1

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

$\frac{1}{1 - e^{\frac{lim_{x \to 1^{+}} x + 1}{lim_{x \to 1^{+}} x + 3}}}$

Step 10.2

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

$\frac{1}{1 - e^{\frac{1 + 1}{lim_{x \to 1^{+}} x + 3}}}$

Step 10.3

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

$\frac{1}{1 - e^{\frac{1 + 1}{1 + 3}}}$

Step 11

Simplify the answer.

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Step 11.1

Add $1$ and $1$ .

$\frac{1}{1 - e^{\frac{2}{1 + 3}}}$

Step 11.2

Add $1$ and $3$ .

$\frac{1}{1 - e^{\frac{2}{4}}}$

Step 11.3

Simplify the denominator.

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Step 11.3.1

Rewrite $1$ as $1^{2}$ .

$\frac{1}{1^{2} - e^{\frac{2}{4}}}$

Step 11.3.2

Rewrite $e^{\frac{2}{4}}$ as ${(e^{\frac{1}{4}})}^{2}$ .

$\frac{1}{1^{2} - {(e^{\frac{1}{4}})}^{2}}$

Step 11.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e^{\frac{1}{4}}$ .

$\frac{1}{(1 + e^{\frac{1}{4}}) (1 - e^{\frac{1}{4}})}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{(1 + e^{\frac{1}{4}}) (1 - e^{\frac{1}{4}})}$

Decimal Form:

$- 1.54149408 \dots$