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Question 1179801: Arrange the functions for which the result is a non-infinite value and the limit exists in ascending order of their limit values as x tends to infinity.
Only 6 functions will be used
f(x)=x^2-1,000/x-5
j(x)=x^2-1/|7x-1|
h(x)=x^3-x^2+4/1-3x^3
i(x)=x-1/|1-4x|
k(x)=5x+1,000/x^2
m(x)=- 4x^2-6/1-4x^2
g(x)=|4x-1|/x-4
l(x)=5x^2-4/x^2+1
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's the analysis of the functions and their limits as x approaches infinity:
**1. Analyze the functions:**
* **f(x) = (x² - 1000) / (x - 5):**
* As x → ∞, the x² term dominates, and the function behaves like x²/x = x.
* Therefore, the limit is ∞ (infinite).
* **j(x) = (x² - 1) / |7x - 1|:**
* As x → ∞, the x² term in the numerator and the 7x term in the denominator dominate.
* The function behaves like x²/7x = x/7.
* Therefore, the limit is ∞ (infinite).
* **h(x) = (x³ - x² + 4) / (1 - 3x³):**
* As x → ∞, the x³ terms dominate.
* The function behaves like x³ / (-3x³) = -1/3.
* Therefore, the limit is -1/3.
* **i(x) = (x - 1) / |1 - 4x|:**
* As x → ∞, the x terms dominate.
* The function behaves like x / |-4x| = x / 4x = 1/4.
* Therefore, the limit is 1/4.
* **k(x) = (5x + 1000) / x²:**
* As x → ∞, the x² term in the denominator dominates.
* The function behaves like 5x / x² = 5/x.
* Therefore, the limit is 0.
* **m(x) = (-4x² - 6) / (1 - 4x²):**
* As x → ∞, the x² terms dominate.
* The function behaves like -4x² / (-4x²) = 1.
* Therefore, the limit is 1.
* **g(x) = |4x - 1| / (x - 4):**
* As x → ∞, the 4x term in the numerator and the x term in the denominator dominate.
* The function behaves like 4x / x = 4.
* Therefore, the limit is 4.
* **l(x) = (5x² - 4) / (x² + 1):**
* As x → ∞, the x² terms dominate.
* The function behaves like 5x² / x² = 5.
* Therefore, the limit is 5.
**2. Identify the functions with non-infinite limits:**
The functions with non-infinite limits are h(x), i(x), k(x), m(x), g(x), and l(x).
**3. Arrange in ascending order:**
* h(x) = -1/3 ≈ -0.333
* k(x) = 0
* i(x) = 1/4 = 0.25
* m(x) = 1
* g(x) = 4
* l(x) = 5
**Final Answer:**
The functions in ascending order of their limits as x tends to infinity are: h(x), k(x), i(x), m(x), g(x), l(x).
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