Math Problem Statement

Проверь пожалйста

lim (x -> 1/2) (8x^3 - 1) / (6x^2 - 5x + 1)

Solution

Let me analyze the solution in the uploaded image and verify the calculations step by step.

Given Problem:

limx128x316x25x+1\lim_{x \to \frac{1}{2}} \frac{8x^3 - 1}{6x^2 - 5x + 1}

From the solution shown:

  1. Initially, substituting x=12x = \frac{1}{2} into the expression results in an indeterminate form 00\frac{0}{0}.
  2. To resolve this, polynomial division and simplification have been used.

Step-by-Step Verification:

Step 1: Substitute x=12x = \frac{1}{2}

We calculate numerator and denominator separately:

  • Numerator: 8x318x^3 - 1 8(12)31=8181=11=08\left(\frac{1}{2}\right)^3 - 1 = 8 \cdot \frac{1}{8} - 1 = 1 - 1 = 0
  • Denominator: 6x25x+16x^2 - 5x + 1 6(12)25(12)+1=61452+1=3252+1=06\left(\frac{1}{2}\right)^2 - 5\left(\frac{1}{2}\right) + 1 = 6 \cdot \frac{1}{4} - \frac{5}{2} + 1 = \frac{3}{2} - \frac{5}{2} + 1 = 0 Thus, substituting x=12x = \frac{1}{2} gives 00\frac{0}{0}, an indeterminate form.

Step 2: Simplify Using Polynomial Division

Perform the polynomial division of: 8x316x25x+1\frac{8x^3 - 1}{6x^2 - 5x + 1}

The solution shows the division step-by-step:

  1. Divide the leading terms:

    • Divide 8x38x^3 by 6x26x^2: 8x6=4x3\frac{8x}{6} = \frac{4x}{3}.
    • Multiply 4x3(6x25x+1)\frac{4x}{3}(6x^2 - 5x + 1), and subtract from the numerator.
  2. Result of division: After dividing, the simplified result appears to be (4x22x+1)/(3x1)(4x^2 - 2x + 1) / (3x - 1).


Step 3: Simplify the Limit Expression

The solution then simplifies: limx124x22x+13x1\lim_{x \to \frac{1}{2}} \frac{4x^2 - 2x + 1}{3x - 1}

Substitute x=12x = \frac{1}{2} into the simplified expression:

  • Numerator: 4x22x+14x^2 - 2x + 1 4(12)22(12)+1=4141+1=14\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) + 1 = 4 \cdot \frac{1}{4} - 1 + 1 = 1
  • Denominator: 3x13x - 1 3(12)1=321=123\left(\frac{1}{2}\right) - 1 = \frac{3}{2} - 1 = \frac{1}{2}

Thus, the limit is: 112=2\frac{1}{\frac{1}{2}} = 2


Final Answer:

The limit is correctly calculated as: 2\boxed{2}


Do you have any questions? Would you like a detailed explanation of any specific step?

Here are 5 related questions to expand your understanding:

  1. How is polynomial division used to simplify rational expressions?
  2. Why does substituting directly into the limit sometimes yield indeterminate forms?
  3. What are alternative methods for solving limits aside from polynomial division?
  4. How does the degree of the numerator and denominator affect the behavior of limits?
  5. What are the common cases where L’Hôpital’s Rule is useful in limits?

Tip: Always simplify the numerator and denominator fully before substituting into a limit to avoid unnecessary complications!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Polynomial Division
Simplification

Formulas

(a^3 - b^3) = (a - b)(a^2 + ab + b^2)
Simplification of rational functions

Theorems

Limit theorem for indeterminate forms

Suitable Grade Level

Grade 11-12 (Pre-Calculus/Calculus)