*Notes to a video lecture on http://www.unizor.com*

__Function Limit - Exercise__

(

(

*x→+∞*)Try to do these exercises yourself.

All function limits below are supposed to be calculated as argument

**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.**

*x*In other words, we say that

**as**

*f(x)→L***, if**

*x→+∞***∀**

*ε>0*∃*A*: (*x ≥ A*)⇒|*f(x)−L*| ≤*ε*Find the limits of the following functions as their argument

**infinitely increasing.**

*x*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

**for all natural**

*n+1 ≤ 2*^{n}**and expand it to all real**

*n*

*x ≥ 1**Answer*:

*0*6.

*x*^{2}/3^{x}*Hint*: Prove that

**for all natural**

*(n+1)*^{2}≤ 2^{n}**and expand it to all real**

*n ≥ 6*

*x ≥ 6**Answer*:

*0*
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