Calculus Examples

$lim_{x \to 1} e^{6 x^{2} - 6 x}$

Step 1

Move the limit into the exponent.

$e^{lim_{x \to 1} 6 x^{2} - 6 x}$

Step 2

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

$e^{lim_{x \to 1} 6 x^{2} - lim_{x \to 1} 6 x}$

Step 3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$e^{6 lim_{x \to 1} x^{2} - lim_{x \to 1} 6 x}$

Step 4

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

$e^{6 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 6 x}$

Step 5

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$e^{6 {(lim_{x \to 1} x)}^{2} - 6 lim_{x \to 1} x}$

Step 6

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

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Step 6.1

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

$e^{6 \cdot 1^{2} - 6 lim_{x \to 1} x}$

Step 6.2

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

$e^{6 \cdot 1^{2} - 6 \cdot 1}$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

One to any power is one.

$e^{6 \cdot 1 - 6 \cdot 1}$

Step 7.1.2

Multiply $6$ by $1$ .

$e^{6 - 6 \cdot 1}$

Step 7.1.3

Multiply $- 6$ by $1$ .

$e^{6 - 6}$

Step 7.2

Subtract $6$ from $6$ .

$e^{0}$

Step 7.3

Anything raised to $0$ is $1$ .

$1$