Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Move the exponent $3$ from ${(1 + 4 x^{2} + 3 x^{4})}^{3}$ outside the limit using the Limits Power Rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Multiply $3$ by $1$ .

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

Step 13.1.2

Add $1$ and $3$ .

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

Step 13.2

Simplify the denominator.

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Step 13.2.1

One to any power is one.

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

Step 13.2.2

Multiply $4$ by $1$ .

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

Step 13.2.3

One to any power is one.

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

Step 13.2.4

Multiply $3$ by $1$ .

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

Step 13.2.5

Add $1$ and $4$ .

$\frac{4}{{(5 + 3)}^{3}}$

Step 13.2.6

Add $5$ and $3$ .

$\frac{4}{8^{3}}$

Step 13.2.7

Raise $8$ to the power of $3$ .

$\frac{4}{512}$

Step 13.3

Cancel the common factor of $4$ and $512$ .

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Step 13.3.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{512}$

Step 13.3.2

Cancel the common factors.

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Step 13.3.2.1

Factor $4$ out of $512$ .

$\frac{4 \cdot 1}{4 \cdot 128}$

Step 13.3.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 128}$

Step 13.3.2.3

Rewrite the expression.

$\frac{1}{128}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{128}$

Decimal Form:

$0.0078125$