Math Problem Statement

Find the limit lim_{x \to \infty} \frac{53x^3 - 27x + 95}{48x^3 - 47}

Solution

This document contains three mathematical problems. Below is their breakdown along with solutions:


Problem 1: Find the limit

limx53x327x+9548x347\lim_{x \to \infty} \frac{53x^3 - 27x + 95}{48x^3 - 47}

Solution:

To solve this, divide the numerator and denominator by x3x^3 (the highest power of xx):

53x327x+9548x347=5327x2+95x34847x3\frac{53x^3 - 27x + 95}{48x^3 - 47} = \frac{53 - \frac{27}{x^2} + \frac{95}{x^3}}{48 - \frac{47}{x^3}}

As xx \to \infty, terms involving 1/x1/x approach 0. Thus:

limx53x327x+9548x347=5348\lim_{x \to \infty} \frac{53x^3 - 27x + 95}{48x^3 - 47} = \frac{53}{48}

Answer: 5348\frac{53}{48}


Problem 2: Solve the differential equation

53y(x)48x+y(x)+28=0-53y(x) - 48x + y'(x) + 28 = 0

Solution:

This is a first-order linear differential equation. Rewrite it as:

y(x)=53y(x)+48x28y'(x) = 53y(x) + 48x - 28

Step 1: Solve the homogeneous equation

y(x)53y(x)=0y'(x) - 53y(x) = 0

This has the solution yh(x)=Ce53xy_h(x) = Ce^{53x}, where CC is an arbitrary constant.

Step 2: Find a particular solution

We use the method of undetermined coefficients. Assume yp(x)=Ax+By_p(x) = Ax + B. Substituting into the equation:

yp(x)=Aandyp(x)=Ax+By_p'(x) = A \quad \text{and} \quad y_p(x) = Ax + B

Substituting yp(x)y_p(x) and yp(x)y_p'(x) into the original equation:

A53(Ax+B)=48x+28A - 53(Ax + B) = -48x + 28

Equating coefficients of xx and constants:

  1. Coefficient of xx: 53A=48    A=4853-53A = -48 \implies A = \frac{48}{53}
  2. Constant: A53B=28    485353B=28    B=4853532853A - 53B = 28 \implies \frac{48}{53} - 53B = 28 \implies B = \frac{48}{53 \cdot 53} - \frac{28}{53}

Thus, the general solution is:

y(x) = Ce^{53x} + \frac{48}{53}x + \text{(simplify B) for constant.}


Problem 3: Plot a polar graph

Given polar function:

r(t)=100(2sin(7t)12cos(30t))100+(tπ2)8,t[π2,3π2]r(t) = \frac{100 (2 - \sin(7t) - \frac{1}{2}\cos(30t))}{100 + (t - \frac{\pi}{2})^8}, \quad t \in \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right]

Instructions:

  1. Use PolarPlot in a mathematical plotting software (e.g., Mathematica).
  2. Customize the plot using:
    • Axes -> False
    • PlotStyle -> {Thickness[.007], RGBColor[0, 1, 0]}

Would you like a detailed explanation of any step?

Further Questions:

  1. How can the general solution of the differential equation be verified?
  2. What does dividing terms by x3x^3 achieve in Problem 1?
  3. Why does the method of undetermined coefficients work for Problem 2?
  4. How does modifying the plot style in PolarPlot improve visualization?
  5. How would the solution in Problem 1 change if the limit was xx \to -\infty?

Tip:

When solving limits, always simplify by dividing by the highest power of the variable in the denominator to isolate the dominant terms.

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Math Problem Analysis

Mathematical Concepts

Limits
Polynomials
Infinity

Formulas

\lim_{x \to \infty} \frac{f(x)}{g(x)}
\frac{53x^3 - 27x + 95}{48x^3 - 47}

Theorems

Limit of rational functions at infinity

Suitable Grade Level

Grades 11-12