Math Problem Statement
dengan menggunakan limit
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:
-
Numerator Simplification:
The numerator is already simplified. It factors as . -
Vertical Asymptote:
The vertical asymptote occurs where the denominator is zero, i.e., .
To confirm, calculate the limits:
Thus, no vertical asymptote exists, as the limit is finite.
Part (b):
Function:
-
Numerator Simplification:
The numerator factors as . -
Vertical Asymptote:
The vertical asymptote occurs where the denominator is zero, i.e., .
To confirm, calculate the limits:
This confirms a vertical asymptote at .
Results:
- Part (a): No vertical asymptote as the limits are finite at .
- Part (b): A vertical asymptote exists at .
Let me know if you have further questions or want detailed explanations!
5 Related Questions:
- How do horizontal asymptotes differ from vertical asymptotes?
- Can there be multiple vertical asymptotes in a rational function?
- How do you find horizontal asymptotes using limits?
- Why is it important to simplify numerators and denominators before analyzing limits?
- 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
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