Math Problem Statement

Solve the limit problems listed on the whiteboard.

Solution

The image you uploaded contains several limit problems. Let's go through them one by one.

Problem 1:

limx3(x213)\lim_{x \to 3} \left(x^2 - 13\right) This is a direct substitution problem. Substitute x=3x = 3: 3213=913=43^2 - 13 = 9 - 13 = -4 So, the answer is: 4\boxed{-4}

Problem 2:

limx2(x32x2+4x+8)\lim_{x \to 2} \left(x^3 - 2x^2 + 4x + 8\right) Substitute x=2x = 2: 232(22)+4(2)+8=88+8+8=162^3 - 2(2^2) + 4(2) + 8 = 8 - 8 + 8 + 8 = 16 Thus, the answer is: 16\boxed{16}

Problem 3:

lims7/3(83s)(2s1)\lim_{s \to 7/3} \left(8 - 3s\right)(2s - 1) Substitute s=73s = \frac{7}{3}: (83(73))(2(73)1)=(87)(1431)=1(14333)=113\left(8 - 3\left(\frac{7}{3}\right)\right)\left(2\left(\frac{7}{3}\right) - 1\right) = \left(8 - 7\right)\left(\frac{14}{3} - 1\right) = 1\left(\frac{14}{3} - \frac{3}{3}\right) = \frac{11}{3} So, the answer is: 113\boxed{\frac{11}{3}}

Problem 4:

limh2(h2+5h2)\lim_{h \to 2} \left(h^2 + 5h - 2\right) Substitute h=2h = 2: 22+5(2)2=4+102=122^2 + 5(2) - 2 = 4 + 10 - 2 = 12 Thus, the answer is: 12\boxed{12}

Problem 5:

limx2(x24)\lim_{x \to 2} \left(x^2 - 4\right) Substitute x=2x = 2: 224=44=02^2 - 4 = 4 - 4 = 0 The answer is: 0\boxed{0}

Would you like to continue with the other problems? Let me know if you want further details on any step.


Here are some related questions:

  1. How do you solve limits that result in indeterminate forms like 00\frac{0}{0}?
  2. What is the difference between one-sided and two-sided limits?
  3. How can L'Hopital's rule help with solving more complex limit problems?
  4. What are some common techniques for solving polynomial limits?
  5. How do you compute limits involving trigonometric functions?

Tip: When direct substitution in a limit gives 00\frac{0}{0}, try factoring or applying L'Hopital's rule to resolve the indeterminate form.

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Math Problem Analysis

Mathematical Concepts

Limits
Substitution in Limits
Factoring
Indeterminate Forms

Formulas

Limit as x approaches a value: lim(x→c) f(x)
Direct substitution
Factoring to simplify expressions

Theorems

Limit Laws
L'Hopital's Rule for indeterminate forms

Suitable Grade Level

Grades 11-12 (Pre-Calculus and Calculus)