Calculus Examples

$lim_{x \to - 8} e^{2 x} - e^{x} - 2$

Step 1

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

$lim_{x \to - 8} e^{2 x} - lim_{x \to - 8} e^{x} - lim_{x \to - 8} 2$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to - 8} 2 x} - lim_{x \to - 8} e^{x} - lim_{x \to - 8} 2$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$e^{2 lim_{x \to - 8} x} - lim_{x \to - 8} e^{x} - lim_{x \to - 8} 2$

Step 4

Move the limit into the exponent.

$e^{2 lim_{x \to - 8} x} - e^{lim_{x \to - 8} x} - lim_{x \to - 8} 2$

Step 5

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

$e^{2 lim_{x \to - 8} x} - e^{lim_{x \to - 8} x} - 1 \cdot 2$

Step 6

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

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Step 6.1

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

$e^{2 \cdot - 8} - e^{lim_{x \to - 8} x} - 1 \cdot 2$

Step 6.2

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

$e^{2 \cdot - 8} - e^{- 8} - 1 \cdot 2$

Step 7

Simplify each term.

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Step 7.1

Multiply $2$ by $- 8$ .

$e^{- 16} - e^{- 8} - 1 \cdot 2$

Step 7.2

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

$\frac{1}{e^{16}} - e^{- 8} - 1 \cdot 2$

Step 7.3

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

$\frac{1}{e^{16}} - \frac{1}{e^{8}} - 1 \cdot 2$

Step 7.4

Multiply $- 1$ by $2$ .

$\frac{1}{e^{16}} - \frac{1}{e^{8}} - 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e^{16}} - \frac{1}{e^{8}} - 2$

Decimal Form:

$- 2.00033535 \dots$