Calculus Examples

$\frac{lim_{x \to 1} 1 + 3 x^{3}}{{(1 + 4 x^{2} + 3 x^{4})}^{3}}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$\frac{lim_{x \to 1} 1 + lim_{x \to 1} 3 x^{3}}{{(1 + 4 x^{2} + 3 x^{4})}^{3}}$

Step 1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{1 + lim_{x \to 1} 3 x^{3}}{{(1 + 4 x^{2} + 3 x^{4})}^{3}}$

Step 1.3

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

$\frac{1 + 3 lim_{x \to 1} x^{3}}{{(1 + 4 x^{2} + 3 x^{4})}^{3}}$

Step 1.4

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{1 + 3 {(lim_{x \to 1} x)}^{3}}{{(1 + 4 x^{2} + 3 x^{4})}^{3}}$

Step 2

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

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

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

One to any power is one.

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

Step 3.1.2

Multiply $3$ by $1$ .

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

Step 3.1.3

Add $1$ and $3$ .

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

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Factor by grouping.

Tap for more steps...

Step 3.2.1.1

Reorder terms.

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

Step 3.2.1.2

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 $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = 4$ .

Tap for more steps...

Step 3.2.1.2.1

Factor $4$ out of $4 x^{2}$ .

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

Step 3.2.1.2.2

Rewrite $4$ as $1$ plus $3$

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

Step 3.2.1.2.3

Apply the distributive property.

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

Step 3.2.1.2.4

Multiply $x^{2}$ by $1$ .

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

Step 3.2.1.3

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1.3.1

Group the first two terms and the last two terms.

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

Step 3.2.1.3.2

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

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

Step 3.2.1.4

Factor the polynomial by factoring out the greatest common factor, $3 x^{2} + 1$ .

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

Step 3.2.2

Apply the product rule to $(3 x^{2} + 1) (x^{2} + 1)$ .

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