Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Cancel the common factor of $x^{3}$ .

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Step 1.1.1

Cancel the common factor.

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

Step 1.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{4}{x} - \frac{3}{x^{2}}}{1}$

Step 1.2

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

$\frac{lim_{x \to \infty} \frac{4}{x} - \frac{3}{x^{2}}}{lim_{x \to \infty} 1}$

Step 1.3

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

$\frac{lim_{x \to \infty} \frac{4}{x} - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1}$

Step 1.4

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

$\frac{4 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1}$

Step 2

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

$\frac{4 \cdot 0 - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1}$

Step 3

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

$\frac{4 \cdot 0 - 3 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 1}$

Step 4

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

$\frac{4 \cdot 0 - 3 \cdot 0}{lim_{x \to \infty} 1}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{4 \cdot 0 - 3 \cdot 0}{1}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Divide $4 \cdot 0 - 3 \cdot 0$ by $1$ .

$4 \cdot 0 - 3 \cdot 0$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply $4$ by $0$ .

$0 - 3 \cdot 0$

Step 5.2.2.2

Multiply $- 3$ by $0$ .

$0 + 0$

Step 5.2.3

Add $0$ and $0$ .

$0$