Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

$\frac{{(lim_{x \to 8} x)}^{2} - lim_{x \to 8} x^{2} \cdot lim_{x \to 8} e^{x^{2}}}{e^{x^{2}}}$

Step 4

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

$\frac{{(lim_{x \to 8} x)}^{2} - {(lim_{x \to 8} x)}^{2} \cdot lim_{x \to 8} e^{x^{2}}}{e^{x^{2}}}$

Step 5

Move the limit into the exponent.

$\frac{{(lim_{x \to 8} x)}^{2} - {(lim_{x \to 8} x)}^{2} \cdot e^{lim_{x \to 8} x^{2}}}{e^{x^{2}}}$

Step 6

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

$\frac{{(lim_{x \to 8} x)}^{2} - {(lim_{x \to 8} x)}^{2} \cdot e^{{(lim_{x \to 8} x)}^{2}}}{e^{x^{2}}}$

Step 7

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

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Step 7.1

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

$\frac{8^{2} - {(lim_{x \to 8} x)}^{2} \cdot e^{{(lim_{x \to 8} x)}^{2}}}{e^{x^{2}}}$

Step 7.2

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

$\frac{8^{2} - 8^{2} \cdot e^{{(lim_{x \to 8} x)}^{2}}}{e^{x^{2}}}$

Step 7.3

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

$\frac{8^{2} - 8^{2} \cdot e^{8^{2}}}{e^{x^{2}}}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $8$ to the power of $2$ .

$\frac{64 - 8^{2} \cdot e^{8^{2}}}{e^{x^{2}}}$

Step 8.2

Raise $8$ to the power of $2$ .

$\frac{64 - 1 \cdot 64 \cdot e^{8^{2}}}{e^{x^{2}}}$

Step 8.3

Multiply $- 1$ by $64$ .

$\frac{64 - 64 \cdot e^{8^{2}}}{e^{x^{2}}}$

Step 8.4

Raise $8$ to the power of $2$ .

$\frac{64 - 64 \cdot e^{64}}{e^{x^{2}}}$

Step 8.5

Rewrite $64 - 64 e^{64}$ in a factored form.

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Step 8.5.1

Rewrite $64$ as $8^{2}$ .

$\frac{8^{2} - 64 e^{64}}{e^{x^{2}}}$

Step 8.5.2

Rewrite $64 e^{64}$ as ${(8 e^{32})}^{2}$ .

$\frac{8^{2} - {(8 e^{32})}^{2}}{e^{x^{2}}}$

Step 8.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = 8 e^{32}$ .

$\frac{(8 + 8 e^{32}) (8 - (8 e^{32}))}{e^{x^{2}}}$

Step 8.5.4

Multiply $8$ by $- 1$ .

$\frac{(8 + 8 e^{32}) (8 - 8 e^{32})}{e^{x^{2}}}$