Calculus Examples

$lim_{x \to 1} e^{- 9 x^{5} - 4 x^{4} - 3 x^{2}}$

Step 1

Move the limit into the exponent.

$e^{lim_{x \to 1} - 9 x^{5} - 4 x^{4} - 3 x^{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$e^{- lim_{x \to 1} 9 x^{5} - lim_{x \to 1} 4 x^{4} - lim_{x \to 1} 3 x^{2}}$

Step 3

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$e^{- 9 lim_{x \to 1} x^{5} - lim_{x \to 1} 4 x^{4} - lim_{x \to 1} 3 x^{2}}$

Step 4

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$e^{- 9 {(lim_{x \to 1} x)}^{5} - lim_{x \to 1} 4 x^{4} - lim_{x \to 1} 3 x^{2}}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$e^{- 9 {(lim_{x \to 1} x)}^{5} - 4 lim_{x \to 1} x^{4} - lim_{x \to 1} 3 x^{2}}$

Step 6

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$e^{- 9 {(lim_{x \to 1} x)}^{5} - 4 {(lim_{x \to 1} x)}^{4} - lim_{x \to 1} 3 x^{2}}$

Step 7

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$e^{- 9 {(lim_{x \to 1} x)}^{5} - 4 {(lim_{x \to 1} x)}^{4} - 3 lim_{x \to 1} x^{2}}$

Step 8

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$e^{- 9 {(lim_{x \to 1} x)}^{5} - 4 {(lim_{x \to 1} x)}^{4} - 3 {(lim_{x \to 1} x)}^{2}}$

Step 9

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$e^{- 9 \cdot 1^{5} - 4 {(lim_{x \to 1} x)}^{4} - 3 {(lim_{x \to 1} x)}^{2}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$e^{- 9 \cdot 1^{5} - 4 \cdot 1^{4} - 3 {(lim_{x \to 1} x)}^{2}}$

Step 9.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$e^{- 9 \cdot 1^{5} - 4 \cdot 1^{4} - 3 \cdot 1^{2}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

One to any power is one.

$e^{- 9 \cdot 1 - 4 \cdot 1^{4} - 3 \cdot 1^{2}}$

Step 10.1.2

Multiply $- 9$ by $1$ .

$e^{- 9 - 4 \cdot 1^{4} - 3 \cdot 1^{2}}$

Step 10.1.3

One to any power is one.

$e^{- 9 - 4 \cdot 1 - 3 \cdot 1^{2}}$

Step 10.1.4

Multiply $- 4$ by $1$ .

$e^{- 9 - 4 - 3 \cdot 1^{2}}$

Step 10.1.5

One to any power is one.

$e^{- 9 - 4 - 3 \cdot 1}$

Step 10.1.6

Multiply $- 3$ by $1$ .

$e^{- 9 - 4 - 3}$

Step 10.2

Subtract $4$ from $- 9$ .

$e^{- 13 - 3}$

Step 10.3

Subtract $3$ from $- 13$ .

$e^{- 16}$

Step 10.4

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

$\frac{1}{e^{16}}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e^{16}}$

Decimal Form:

$1.12535174 \cdot 10^{- 7}$