A function is defined by for all

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A function

g

is defined by

g(x)=\dfrac{x^{2}-1}{x-1}

for all

x\neq 1

. Numerical approximations of

g(x)

near

x=1

are provided in separate tables for values less than 1 and greater than 1. Left of

x=1

x 0.90 0.99 0.999 g(x) 1.900 1.990 1.999 Right of

x=1

x 1.001 1.01 1.10 g(x) 2.001 2.010 2.100 The value of the function at

x=1

is specified as

g(1)=5

. (a) Use the left–hand data to estimate

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

. Round your answer to three decimal places. (b) Use the right–hand data to estimate

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

. Round your answer to three decimal places. (c) Determine the value of

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

. Provide a brief justification based on your results from parts (a) and (b). (d) Is

g

continuous at

x=1

? Support your response with reference to both the limit you found in part (c) and the given value

g(1)=5