Answer the following questions related to the basic idea of a limit and continuity.

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

Answer the following questions related to the basic idea of a limit and continuity. (i) Define the limit of a function

f(x)

x

approaches a real number

a

. State the meaning of the notation

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

. (ii) State the rule that allows you to evaluate

\displaystyle \lim_{x \to a} f(x)

by simply substituting

x=a

when

f(x)

is a polynomial or a rational function whose denominator is non–zero at

x=a

. (iii) Give two distinct examples of elementary functions that are continuous at every point of their domains and briefly justify your choices.