Answer the following questions related to the idea of a limit

MATHEMATICS · Subjective · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

Answer the following questions related to the idea of a limit. (a) Define the limit of a function

f(x)

x

approaches a real number

a

in intuitive terms. (b) Using an explicit example of a polynomial

P(x)

, explain why

\displaystyle \lim_{x\to a} P(x)=P(a)

whenever

P

is defined at

x=a

. (c) A rational function is given by

R(x)=\dfrac{p(x)}{q(x)}

, where

p

and

q

are polynomials. State one specific condition on

q(x)

that must hold at

x=a

for the limit

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

to be obtained by direct substitution, and justify your answer with a numerical example.