Calculus Examples

$lim_{t \to 8} \frac{1}{1 + a e^{- k t}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{t \to 8} 1}{lim_{t \to 8} 1 + a e^{- k t}}$

Step 1.2

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

$\frac{1}{lim_{t \to 8} 1 + a e^{- k t}}$

Step 1.3

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

$\frac{1}{lim_{t \to 8} 1 + lim_{t \to 8} a e^{- k t}}$

Step 1.4

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

$\frac{1}{1 + lim_{t \to 8} a e^{- k t}}$

Step 1.5

Move the term $a$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{1 + a lim_{t \to 8} e^{- k t}}$

Step 1.6

Move the limit into the exponent.

$\frac{1}{1 + a e^{lim_{t \to 8} - k t}}$

Step 1.7

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

$\frac{1}{1 + a e^{- k lim_{t \to 8} t}}$

Step 2

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

$\frac{1}{1 + a e^{- k \cdot 8}}$

Step 3

Multiply $8$ by $- 1$ .

$\frac{1}{1 + a e^{- 8 k}}$