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Unizor - Function Limits - Exercise 1
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Unizor - Creative Minds through Art of Mathematics - Math4Teens
Function Limit - Exercise
(x→+∞)
Try to do these exercises yourself.
All function limits below are supposed to be calculated as argument x tends to positive infinity, that is infinitely increasing without bounds, eventually getting larger than any real number fixed beforehand and staying larger than that number ever since.
In other words, we say that
f(x)→L as x→+∞, if
∀ (positive) ε ∃A: (x ≥ A)⇒|f(x)−L| ≤ ε
Find the limits of the following functions as their argument x infinitely increasing.
1. (2x²+3x+4)/(3x²+4x+5)
Answer: 2/3
2. (2x³+3x+4)/(3x²+4x+5)
Answer: Function will be infinitely increasing or, non-rigorously, its limit is +∞
3. (2x²+3x+4)/(3x³+4x+5)
Answer: 0
4. x·sin(1/x)
Answer: 1
5. x/(3^x)
Hint: Prove that n+1 ≤ 2^n for all natural n and expand it to all real x ≥ 1
Answer: 0
6. (x^2)/(3^x)
Hint: Prove that (n+1)^2 ≤ 2^n for all natural n ≥ 6 and expand it to all real x ≥ 6
Answer: 0
Function Limit - Exercise
(x→+∞)
Try to do these exercises yourself.
All function limits below are supposed to be calculated as argument x tends to positive infinity, that is infinitely increasing without bounds, eventually getting larger than any real number fixed beforehand and staying larger than that number ever since.
In other words, we say that
f(x)→L as x→+∞, if
∀ (positive) ε ∃A: (x ≥ A)⇒|f(x)−L| ≤ ε
Find the limits of the following functions as their argument x infinitely increasing.
1. (2x²+3x+4)/(3x²+4x+5)
Answer: 2/3
2. (2x³+3x+4)/(3x²+4x+5)
Answer: Function will be infinitely increasing or, non-rigorously, its limit is +∞
3. (2x²+3x+4)/(3x³+4x+5)
Answer: 0
4. x·sin(1/x)
Answer: 1
5. x/(3^x)
Hint: Prove that n+1 ≤ 2^n for all natural n and expand it to all real x ≥ 1
Answer: 0
6. (x^2)/(3^x)
Hint: Prove that (n+1)^2 ≤ 2^n for all natural n ≥ 6 and expand it to all real x ≥ 6
Answer: 0
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