A piecewise–defined function is tabulated for –values near

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

Problem

A piecewise–defined function

h

is tabulated for

x

–values near

x=5

$x$	4.8	4.9	4.99	5.01	5.1	5.2
$h(x)$	7.2	7.4	7.9	2.1	2.05	2.0

(a) Estimate

h(5^-)

and

h(5^+)

, the values that

h(x)

appears to approach as

x

approaches

5

from the left and from the right, respectively, and clearly state your reasoning. (b) Determine the value of

\displaystyle \lim_{x\to 5} h(x)

, if it exists. Provide a brief justification grounded in the estimates from part (a). (c) Suppose a new function

k

is defined by keeping

k(x)=h(x)

for

x\neq 5

and setting

k(5)=4.75

. Would

k

be continuous at

x=5

? Explain fully.