Consider the concept of the limit of a function at infinity. Answer the following questions.
a. (i) State the formal ε–M definition of x→∞limf(x)=L.
b. (ii) For f(x)=x1, use the definition stated in part (i) to give a clear verbal explanation (no detailed algebraic proof required) of why x→∞limx1=0.
c. (iii) A student writes the following argument:
“Given ε>0, choose M=ε. If x>M, then
x1−0=x1<M1=ε1<ε.
Hence the limit is proved.”
Identify the logical error in this reasoning and briefly indicate how it should be corrected.