Calculus Examples

$lim_{x \to \infty} \frac{4 x^{4} - 49}{(x^{2} + 49) (4 x^{2} - 49)}$

Step 1

Simplify.

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Step 1.1

Expand $(x^{2} + 49) (4 x^{2} - 49)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$lim_{x \to \infty} \frac{4 x^{4} - 49}{x^{2} (4 x^{2} - 49) + 49 (4 x^{2} - 49)}$

Step 1.1.2

Apply the distributive property.

$lim_{x \to \infty} \frac{4 x^{4} - 49}{x^{2} (4 x^{2}) + x^{2} \cdot - 49 + 49 (4 x^{2} - 49)}$

Step 1.1.3

Apply the distributive property.

$lim_{x \to \infty} \frac{4 x^{4} - 49}{x^{2} (4 x^{2}) + x^{2} \cdot - 49 + 49 (4 x^{2}) + 49 \cdot - 49}$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 x^{2} x^{2} + x^{2} \cdot - 49 + 49 (4 x^{2}) + 49 \cdot - 49}$

Step 1.2.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.2.1.2.1

Move $x^{2}$ .

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 (x^{2} x^{2}) + x^{2} \cdot - 49 + 49 (4 x^{2}) + 49 \cdot - 49}$

Step 1.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 x^{2 + 2} + x^{2} \cdot - 49 + 49 (4 x^{2}) + 49 \cdot - 49}$

Step 1.2.1.2.3

Add $2$ and $2$ .

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 x^{4} + x^{2} \cdot - 49 + 49 (4 x^{2}) + 49 \cdot - 49}$

Step 1.2.1.3

Move $- 49$ to the left of $x^{2}$ .

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 x^{4} - 49 \cdot x^{2} + 49 (4 x^{2}) + 49 \cdot - 49}$

Step 1.2.1.4

Multiply $4$ by $49$ .

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 x^{4} - 49 x^{2} + 196 x^{2} + 49 \cdot - 49}$

Step 1.2.1.5

Multiply $49$ by $- 49$ .

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 x^{4} - 49 x^{2} + 196 x^{2} - 2401}$

Step 1.2.2

Add $- 49 x^{2}$ and $196 x^{2}$ .

$lim_{x \to \infty} \frac{4 x^{4} - 49}{4 x^{4} + 147 x^{2} - 2401}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{4 x^{4}}{x^{4}} + \frac{- 49}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{147 x^{2}}{x^{4}} + \frac{- 2401}{x^{4}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

Cancel the common factor of $x^{4}$ .

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Step 3.1.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{4 x^{4}}{x^{4}} + \frac{- 49}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{147 x^{2}}{x^{4}} + \frac{- 2401}{x^{4}}}$

Step 3.1.1.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{4 + \frac{- 49}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{147 x^{2}}{x^{4}} + \frac{- 2401}{x^{4}}}$

Step 3.1.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{147 x^{2}}{x^{4}} + \frac{- 2401}{x^{4}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $x^{4}$ .

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Step 3.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{147 x^{2}}{x^{4}} + \frac{- 2401}{x^{4}}}$

Step 3.2.1.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{4 + \frac{147 x^{2}}{x^{4}} + \frac{- 2401}{x^{4}}}$

Step 3.2.2

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 3.2.2.1

Factor $x^{2}$ out of $147 x^{2}$ .

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{4 + \frac{x^{2} \cdot 147}{x^{4}} + \frac{- 2401}{x^{4}}}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

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

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{4 + \frac{x^{2} \cdot 147}{x^{2} x^{2}} + \frac{- 2401}{x^{4}}}$

Step 3.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{4 + \frac{x^{2} \cdot 147}{x^{2} x^{2}} + \frac{- 2401}{x^{4}}}$

Step 3.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{4 + \frac{147}{x^{2}} + \frac{- 2401}{x^{4}}}$

Step 3.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{4 - \frac{49}{x^{4}}}{4 + \frac{147}{x^{2}} - \frac{2401}{x^{4}}}$

Step 3.3

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

$\frac{lim_{x \to \infty} 4 - \frac{49}{x^{4}}}{lim_{x \to \infty} 4 + \frac{147}{x^{2}} - \frac{2401}{x^{4}}}$

Step 3.4

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

$\frac{lim_{x \to \infty} 4 - lim_{x \to \infty} \frac{49}{x^{4}}}{lim_{x \to \infty} 4 + \frac{147}{x^{2}} - \frac{2401}{x^{4}}}$

Step 3.5

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{4 - lim_{x \to \infty} \frac{49}{x^{4}}}{lim_{x \to \infty} 4 + \frac{147}{x^{2}} - \frac{2401}{x^{4}}}$

Step 3.6

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 - 49 lim_{x \to \infty} \frac{1}{x^{4}}}{lim_{x \to \infty} 4 + \frac{147}{x^{2}} - \frac{2401}{x^{4}}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

$\frac{4 - 49 \cdot 0}{lim_{x \to \infty} 4 + \frac{147}{x^{2}} - \frac{2401}{x^{4}}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{4 - 49 \cdot 0}{lim_{x \to \infty} 4 + lim_{x \to \infty} \frac{147}{x^{2}} - lim_{x \to \infty} \frac{2401}{x^{4}}}$

Step 5.2

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{4 - 49 \cdot 0}{4 + lim_{x \to \infty} \frac{147}{x^{2}} - lim_{x \to \infty} \frac{2401}{x^{4}}}$

Step 5.3

Move the term $147$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 - 49 \cdot 0}{4 + 147 lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} \frac{2401}{x^{4}}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{4 - 49 \cdot 0}{4 + 147 \cdot 0 - lim_{x \to \infty} \frac{2401}{x^{4}}}$

Step 7

Move the term $2401$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 - 49 \cdot 0}{4 + 147 \cdot 0 - 2401 lim_{x \to \infty} \frac{1}{x^{4}}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

$\frac{4 - 49 \cdot 0}{4 + 147 \cdot 0 - 2401 \cdot 0}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 49$ by $0$ .

$\frac{4 + 0}{4 + 147 \cdot 0 - 2401 \cdot 0}$

Step 9.1.2

Add $4$ and $0$ .

$\frac{4}{4 + 147 \cdot 0 - 2401 \cdot 0}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $147$ by $0$ .

$\frac{4}{4 + 0 - 2401 \cdot 0}$

Step 9.2.2

Multiply $- 2401$ by $0$ .

$\frac{4}{4 + 0 + 0}$

Step 9.2.3

Add $4 + 0$ and $0$ .

$\frac{4}{4 + 0}$

Step 9.2.4

Add $4$ and $0$ .

$\frac{4}{4}$

Step 9.3

Divide $4$ by $4$ .

$1$