lim x approaches to 0 (gx)/f(x)

Question

 

PLEASE ANSWER ASAP. Thank you!

Answer 1

​a) f(0)=1 as x→0 because x<1, g(0)→0 as x→0 so limit of g(x)/f(x)=0/1=0.

b) f(2)→x as x→2 so f(2)→2 because 1<2, g(2)→x because x≠2, therefore g(2)→2, so limit of f(x)/g(x)=2/2=1.

Comments

Hi. I'd like to ask on why g(2) is --> x when clearly, it's equal to 2. I answered it with 2/3 and it was found wrong, but I don't understand why this is so.

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Yes, that’s the trap they want you to fall into! You need to distinguish between x=2 and x approaches 2. x approaches 2 means that x is either less than 2 or greater than 2. So it EXCLUDES the one single point where x=2. You have to apply the condition x is not equal to 2. The purpose of this question is to test that you have a clear understanding of limits. It’s important when a function contains discontinuities. For example f(x)=x/x is 1  except when x=0, when the function is not defined. As x approaches 0 f(x)=1, but at the very point where x=0 it’s undefined. 

Answer 2

f(x) = 3x2 + 4                 

 g(x) = 5x3 + 2