Calculus Examples

$\frac{lim_{x \to 3} x^{4 - 81}}{2 x^{2} - 5 x - 3}$

Step 1

Move the exponent $4 - 81$ from $x^{4 - 81}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 3} x)}^{4 - 81}}{2 x^{2} - 5 x - 3}$

Step 2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{4 - 81}}{2 x^{2} - 5 x - 3}$

Step 3

Simplify the answer.

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Step 3.1

Move $3^{4 - 81}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{(2 x^{2} - 5 x - 3) \cdot 3^{- (4 - 81)}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Factor by grouping.

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Step 3.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

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Step 3.2.1.1.1

Factor $- 5$ out of $- 5 x$ .

$\frac{1}{(2 x^{2} - 5 (x) - 3) \cdot 3^{- (4 - 81)}}$

Step 3.2.1.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{1}{(2 x^{2} + (1 - 6) x - 3) \cdot 3^{- (4 - 81)}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{1}{(2 x^{2} + 1 x - 6 x - 3) \cdot 3^{- (4 - 81)}}$

Step 3.2.1.1.4

Multiply $x$ by $1$ .

$\frac{1}{(2 x^{2} + x - 6 x - 3) \cdot 3^{- (4 - 81)}}$

Step 3.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.1.2.1

Group the first two terms and the last two terms.

$\frac{1}{((2 x^{2} + x) - 6 x - 3) \cdot 3^{- (4 - 81)}}$

Step 3.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{1}{(x (2 x + 1) - 3 (2 x + 1)) \cdot 3^{- (4 - 81)}}$

Step 3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{1}{(2 x + 1) (x - 3) \cdot 3^{- (4 - 81)}}$

Step 3.2.2

Subtract $81$ from $4$ .

$\frac{1}{(2 x + 1) (x - 3) \cdot 3^{- - 77}}$

Step 3.2.3

Multiply $- 1$ by $- 77$ .

$\frac{1}{(2 x + 1) (x - 3) \cdot 3^{77}}$

Step 3.3

Reorder factors in $\frac{1}{(2 x + 1) (x - 3) \cdot 3^{77}}$ .

$\frac{1}{3^{77} (2 x + 1) (x - 3)}$