Mathematics

Problem

Consider the polynomial function

f(x)=4x^{3}-7x+2

. Answer the following questions related to the limit of

f

x=1

. a. (i) State the definition of the limit

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

in terms of the left–hand limit and the right–hand limit. b. (ii) Calculate the left–hand limit

\displaystyle \lim_{x\to 1^{-}}f(x)

and the right–hand limit

\displaystyle \lim_{x\to 1^{+}}f(x)

. c. (iii) Using your results from part (ii), explain why

\displaystyle \lim_{x\to 1}f(x)

exists and find its value. d. (iv) Verify whether

f

is continuous at

x=1

and justify your answer briefly.