Calculus Examples

$lim_{z \to 8} e^{- z} + \frac{5}{z}$

Step 1

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

$lim_{z \to 8} e^{- z} + lim_{z \to 8} \frac{5}{z}$

Step 2

Move the limit into the exponent.

$e^{lim_{z \to 8} - z} + lim_{z \to 8} \frac{5}{z}$

Step 3

Move the term $- 1$ outside of the limit because it is constant with respect to $z$ .

$e^{- lim_{z \to 8} z} + lim_{z \to 8} \frac{5}{z}$

Step 4

Move the term $5$ outside of the limit because it is constant with respect to $z$ .

$e^{- lim_{z \to 8} z} + 5 lim_{z \to 8} \frac{1}{z}$

Step 5

Split the limit using the Limits Quotient Rule on the limit as $z$ approaches $8$ .

$e^{- lim_{z \to 8} z} + 5 \frac{lim_{z \to 8} 1}{lim_{z \to 8} z}$

Step 6

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

$e^{- lim_{z \to 8} z} + 5 \frac{1}{lim_{z \to 8} z}$

Step 7

Evaluate the limits by plugging in $8$ for all occurrences of $z$ .

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Step 7.1

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

$e^{- 8} + 5 \frac{1}{lim_{z \to 8} z}$

Step 7.2

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

$e^{- 8} + 5 (\frac{1}{8})$

Step 8

Simplify each term.

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Step 8.1

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

$\frac{1}{e^{8}} + 5 (\frac{1}{8})$

Step 8.2

Combine $5$ and $\frac{1}{8}$ .

$\frac{1}{e^{8}} + \frac{5}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e^{8}} + \frac{5}{8}$

Decimal Form:

$0.62533546 \dots$