Calculus Examples

$lim_{x \to 0} e^{x} - \ln (x + 1)$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} e^{x} - lim_{x \to 0} \ln (x + 1)$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 0} x} - lim_{x \to 0} \ln (x + 1)$

Step 3

Move the limit inside the logarithm.

$e^{lim_{x \to 0} x} - \ln (lim_{x \to 0} x + 1)$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$e^{lim_{x \to 0} x} - \ln (lim_{x \to 0} x + lim_{x \to 0} 1)$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$e^{lim_{x \to 0} x} - \ln (lim_{x \to 0} x + 1)$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{0} - \ln (lim_{x \to 0} x + 1)$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{0} - \ln (0 + 1)$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Anything raised to $0$ is $1$ .

$1 - \ln (0 + 1)$

Step 7.1.2

Add $0$ and $1$ .

$1 - \ln (1)$

Step 7.1.3

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

$1 - 0$

Step 7.1.4

Multiply $- 1$ by $0$ .

$1 + 0$

Step 7.2

Add $1$ and $0$ .

$1$