Consider the concept of the limit of a function at infinity

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the concept of the limit of a function at infinity. Answer the following questions. (i) State the formal ε–M definition of

\displaystyle \lim_{x\to\infty}f(x)=L

. (ii) For

f(x)=\dfrac{1}{x}

, use the definition stated in part (i) to give a clear verbal explanation (no detailed algebraic proof required) of why

\displaystyle \lim_{x\to\infty}\frac{1}{x}=0

. (iii) A student writes the following argument: “Given

\varepsilon>0

, choose

M=\varepsilon

. If

x>M

, then

\left|\frac{1}{x}-0\right|=\frac{1}{x}<\frac{1}{M}=\frac{1}{\varepsilon}<\varepsilon.

Hence the limit is proved.” Identify the logical error in this reasoning and briefly indicate how it should be corrected.