Calculus Examples

$lim_{x \to 8} \log (2 + e^{- x + 1})$

Step 1

Move the limit inside the logarithm.

$\log (lim_{x \to 8} 2 + e^{- x + 1})$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\log (lim_{x \to 8} 2 + lim_{x \to 8} e^{- x + 1})$

Step 3

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\log (2 + lim_{x \to 8} e^{- x + 1})$

Step 4

Move the limit into the exponent.

$\log (2 + e^{lim_{x \to 8} - x + 1})$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\log (2 + e^{- lim_{x \to 8} x + lim_{x \to 8} 1})$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\log (2 + e^{- lim_{x \to 8} x + 1})$

Step 7

Simplify terms.

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Step 7.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\log (2 + e^{- 8 + 1})$

Step 7.2

Simplify each term.

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Step 7.2.1

Add $- 8$ and $1$ .

$\log (2 + e^{- 7})$

Step 7.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log (2 + \frac{1}{e^{7}})$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\log (2 + \frac{1}{e^{7}})$

Decimal Form:

$0.30122796 \dots$