Graph the function f(x)=
2x2−100

x2+2
, look at the graph in differently sized viewing windows and determine, if possible, lim
x→∞
2x2−100

x2+2
.

Question

Graph the function f(x)=
2x2−100

x2+2
, look at the graph in differently sized viewing windows and determine, if possible, lim
x→∞
2x2−100

x2+2
.

BlackTom AI · Accepted Answer

First, I’ll restate the problem to ensure clarity: we are asked to evaluate the limit as x approaches infinity of the function (2x^2 - 100) / (x^2 + 2).

Next, I note that there are no answer options provided to analyze, so I will focus on the mathematical reasoning behind the limit.

A common technique for limits of rational functions as x → ∞ is to compare the highest-degree terms in the numerator and the denominator, since they dominate the behavior of the function for large x.

In the expression (2x^2 - 100) / (x^2 + 2), the highest-degree terms are 2x^2 in the numerator and x^2 in the denominator.

If we factor out x^2 from numerator and denominator, we get:
- Numerator: 2x^2 - 100 = x^2(2 - 100/x^2)
- Denominator: x^2 + 2 = x^2(1 + 2/x^2)

Thus the quotient becomes [x^2(2 - 100/x^2)] / [x^2(1 + 2/x^2)] = [2 - 100/x^2] / [1 + 2/x^2].

As x → ∞, the terms 100/x^2 and 2/x^2 go to 0, so the expression tends to 2/1 = 2.

Therefore, the limit is determined by the dominant leading coefficients of the highest-degree terms, yielding a final tendency toward 2 as x grows without bound.

If one were to consider different viewing windows visually, the same horizontal asymptotic behavior would be observed, with the graph approaching the line y = 2 for large |x|, regardless of moderate window sizes.

Graph the function f(x)= 2x2−100 x2+2 , look at the graph in differently sized viewing windows and determine, if possible, lim x→∞ 2x2−100 x2+2 .Single choice

Similar Questions

What is the following limit?

Compute \(\displaystyle \lim _{x\to -\infty }{\frac {x^4-8x^3-3x+5}{6x^4+7x^2+3}}\). (Use I to stand for \(\infty \) if needed.)

Compute [math: limx→−∞x4−8x3−3x+56x4+7x2+3]\displaystyle \lim _{x\to -\infty }{\frac {x^4-8x^3-3x+5}{6x^4+7x^2+3}}. (Use I to stand for [math: ∞]\infty if needed.)

Compute [math: limx→−∞−4x3+2x2+6x−57x4+4x2+1]\displaystyle \lim _{x\to -\infty }{\frac {-4x^3+2x^2+6x-5}{7x^4+4x^2+1}}. (Use I to stand for [math: ∞]\infty if needed.)

Compute [math: limx→−∞x7]\displaystyle \lim _{x\to -\infty }{x^7}. (Use I to stand for [math: ∞]\infty if needed.)

Compute \(\displaystyle \lim _{x\to -\infty }{\frac {3x^4-4x^3-2x+1}{8x^4+9x^2+5}}\). (Use I to stand for \(\infty \) if needed.)

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler