Limits at Infinity — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Evaluate the limit at infinity.

   $$\underset{x\to \infty }{\lim }\frac{3x+1}{x-2}$$

   Answer: $3$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to \infty }{\lim }\frac{3x+1}{x-2}=3$$
2. 2.

   Evaluate the limit at infinity.

   $$\underset{x\to \infty }{\lim }\frac{2x^2-1}{5x^2+3}$$

   Answer: $\frac{2}{5}$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to \infty }{\lim }\frac{2x^2-1}{5x^2+3}=\frac{2}{5}$$
3. 3.

   Evaluate the limit at infinity.

   $$\underset{x\to \infty }{\lim }\frac{7x^3+x}{2x^3-4}$$

   Answer: $\frac{7}{2}$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to \infty }{\lim }\frac{7x^3+x}{2x^3-4}=\frac{7}{2}$$
4. 4.

   Evaluate the limit at infinity.

   $$\underset{x\to \infty }{\lim }\frac{4x^2+x}{x^3+1}$$

   Answer: $0$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to \infty }{\lim }\frac{4x^2+x}{x^3+1}=0$$
5. 5.

   Evaluate the limit at infinity.

   $$\underset{x\to -\infty }{\lim }\frac{5x-2}{2x+7}$$

   Answer: $\frac{5}{2}$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to -\infty }{\lim }\frac{5x-2}{2x+7}=\frac{5}{2}$$
6. 6.

   Evaluate the limit at infinity.

   $$\underset{x\to -\infty }{\lim }\frac{3x^2+8}{2-6x^2}$$

   Answer: $-\frac{1}{2}$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to -\infty }{\lim }\frac{3x^2+8}{2-6x^2}=-\frac{1}{2}$$
7. 7.

   Evaluate the limit at infinity.

   $$\underset{x\to \infty }{\lim }\frac{4x+5}{2x-1}$$

   Answer: $2$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to \infty }{\lim }\frac{4x+5}{2x-1}=2$$
8. 8.

   Evaluate the limit at infinity.

   $$\underset{x\to \infty }{\lim }\frac{6x^2-x}{3x^2+2}$$

   Answer: $2$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to \infty }{\lim }\frac{6x^2-x}{3x^2+2}=2$$
9. 9.

   Evaluate the limit at infinity.

   $$\underset{x\to \infty }{\lim }\frac{x^2+4}{2x^3-1}$$

   Answer: $0$

   Step-by-step solution

   1. $$\text{Compare the highest powers in the numerator and denominator.}$$
   2. $$\underset{x\to \infty }{\lim }\frac{x^2+4}{2x^3-1}=0$$
10. 10.

    Evaluate the limit at infinity.

    $$\underset{x\to -\infty }{\lim }\frac{9x^3-2}{3x^3+x}$$

    Answer: $3$

    Step-by-step solution

    1. $$\text{Compare the highest powers in the numerator and denominator.}$$
    2. $$\underset{x\to -\infty }{\lim }\frac{9x^3-2}{3x^3+x}=3$$
11. 11.

    Evaluate the limit at infinity.

    $$\underset{x\to \infty }{\lim }\frac{2x-9}{7x+4}$$

    Answer: $\frac{2}{7}$

    Step-by-step solution

    1. $$\text{Compare the highest powers in the numerator and denominator.}$$
    2. $$\underset{x\to \infty }{\lim }\frac{2x-9}{7x+4}=\frac{2}{7}$$
12. 12.

    Evaluate the limit at infinity.

    $$\underset{x\to -\infty }{\lim }\frac{5x^2+2x}{x^2-8}$$

    Answer: $5$

    Step-by-step solution

    1. $$\text{Compare the highest powers in the numerator and denominator.}$$
    2. $$\underset{x\to -\infty }{\lim }\frac{5x^2+2x}{x^2-8}=5$$
13. 13.

    Evaluate the limit at infinity.

    $$\underset{x\to \infty }{\lim }\frac{x^4+3}{4x^4-x}$$

    Answer: $\frac{1}{4}$

    Step-by-step solution

    1. $$\text{Compare the highest powers in the numerator and denominator.}$$
    2. $$\underset{x\to \infty }{\lim }\frac{x^4+3}{4x^4-x}=\frac{1}{4}$$
14. 14.

    Evaluate the limit at infinity.

    $$\underset{x\to \infty }{\lim }\frac{8x^3+x^2}{2x^3-5}$$

    Answer: $4$

    Step-by-step solution

    1. $$\text{Compare the highest powers in the numerator and denominator.}$$
    2. $$\underset{x\to \infty }{\lim }\frac{8x^3+x^2}{2x^3-5}=4$$
15. 15.

    Evaluate the limit at infinity.

    $$\underset{x\to -\infty }{\lim }\frac{3x+1}{x^2+6}$$

    Answer: $0$

    Step-by-step solution

    1. $$\text{Compare the highest powers in the numerator and denominator.}$$
    2. $$\underset{x\to -\infty }{\lim }\frac{3x+1}{x^2+6}=0$$
16. 16.

    Evaluate the limit at infinity.

    $$\underset{x\to \infty }{\lim }\frac{10x^2-3}{4-5x^2}$$

    Answer: $-2$

    Step-by-step solution

    1. $$\text{Compare the highest powers in the numerator and denominator.}$$
    2. $$\underset{x\to \infty }{\lim }\frac{10x^2-3}{4-5x^2}=-2$$

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• Evaluating Limits
• One-Sided Limits
• Continuity
• Derivative from First Principles
• Power Rule
• Trigonometric Derivatives
• Exponential & Log Derivatives
• Product Rule
• Quotient Rule
• Chain Rule
• Implicit Differentiation
• Related Rates
• Curve Analysis
• Optimization
• Antiderivatives
• Definite Integrals
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution