Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Move the limit into the exponent.

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

$\frac{1}{1} - \frac{1}{e^{1} - 1 \cdot 1}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Divide $1$ by $1$ .

$1 - \frac{1}{e^{1} - 1 \cdot 1}$

Step 10.1.2

Simplify the denominator.

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Step 10.1.2.1

Simplify.

$1 - \frac{1}{e - 1 \cdot 1}$

Step 10.1.2.2

Multiply $- 1$ by $1$ .

$1 - \frac{1}{e - 1}$

Step 10.2

Write $1$ as a fraction with a common denominator.

$\frac{e - 1}{e - 1} - \frac{1}{e - 1}$

Step 10.3

Combine the numerators over the common denominator.

$\frac{e - 1 - 1}{e - 1}$

Step 10.4

Subtract $1$ from $- 1$ .

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.41802329 \dots$