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.

DIFFERENTIAL CALCULUS (MTH_251_020_W2025) Unit 5 - Discontinuities and Infinite Limits - PreClass

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 .

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