Math Problem Statement

dengan menggunakan limit

Tentukan asimtot tegak fungsi-fungsi berikut, kalau ada: a. f(x) = (x^2 - x - 2) / (x + 1), b. f(x) = (x^2 - x - 2) / (x + 2).

Solution

py import symbols, limit, oo, simplify

Define variables

x = symbols('x')

Define the functions

f_a = (x2 - x - 2) / (x + 1) f_b = (x2 - x - 2) / (x + 2)

Simplify the numerator for root detection

simplified_a = simplify(x2 - x - 2) simplified_b = simplify(x2 - x - 2)

Determine the vertical asymptotes by setting the denominators to zero

asymptote_a = limit(f_a, x, -1, dir='-'), limit(f_a, x, -1, dir='+') asymptote_b = limit(f_b, x, -2, dir='-'), limit(f_b, x, -2, dir='+')

simplified_a, simplified_b, asymptote_a, asymHere are the steps and results for determining the vertical asymptotes using limits:

Part (a):

Function:
f(x)=x2x2x+1f(x) = \frac{x^2 - x - 2}{x + 1}

  1. Numerator Simplification:
    The numerator x2x2x^2 - x - 2 is already simplified. It factors as (x2)(x+1)(x - 2)(x + 1).

  2. Vertical Asymptote:
    The vertical asymptote occurs where the denominator is zero, i.e., x+1=0x=1x + 1 = 0 \rightarrow x = -1.
    To confirm, calculate the limits:
    limx1f(x)=3,limx1+f(x)=3\lim_{x \to -1^-} f(x) = -3, \quad \lim_{x \to -1^+} f(x) = -3 Thus, no vertical asymptote exists, as the limit is finite.


Part (b):

Function:
f(x)=x2x2x+2f(x) = \frac{x^2 - x - 2}{x + 2}

  1. Numerator Simplification:
    The numerator x2x2x^2 - x - 2 factors as (x2)(x+1)(x - 2)(x + 1).

  2. Vertical Asymptote:
    The vertical asymptote occurs where the denominator is zero, i.e., x+2=0x=2x + 2 = 0 \rightarrow x = -2.
    To confirm, calculate the limits:
    limx2f(x)=,limx2+f(x)=\lim_{x \to -2^-} f(x) = -\infty, \quad \lim_{x \to -2^+} f(x) = \infty This confirms a vertical asymptote at x=2x = -2.


Results:

  • Part (a): No vertical asymptote as the limits are finite at x=1x = -1.
  • Part (b): A vertical asymptote exists at x=2x = -2.

Let me know if you have further questions or want detailed explanations!

5 Related Questions:

  1. How do horizontal asymptotes differ from vertical asymptotes?
  2. Can there be multiple vertical asymptotes in a rational function?
  3. How do you find horizontal asymptotes using limits?
  4. Why is it important to simplify numerators and denominators before analyzing limits?
  5. What is the significance of one-sided limits in determining asymptotes?

Tip: Always check both sides of the limit for vertical asymptotes to ensure proper analysis.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Vertical Asymptotes
Limits
Simplification of Polynomials

Formulas

Vertical asymptote occurs where the denominator equals zero
Limit notation: lim(x→a⁻) f(x), lim(x→a⁺) f(x)

Theorems

The behavior of rational functions near vertical asymptotes

Suitable Grade Level

Grades 10-12