Mathematics

Problem

The function

g

is defined for all

x\neq 2

. The table gives values of

g(x)

for

x

close to

2

. x g(x) 1.9 4.02 1.99 4.001 1.999 4.0001 2.001 4.9998 2.01 4.99 2.1 4.90 Which statement best describes the two–sided limit of

g

x

approaches

2

? a) The limit exists and equals

4.5

, the average of the one–sided limits. b) The limit does not exist because values approaching

2

from the left tend toward

4

while values from the right tend toward

5

. c) The limit exists and equals

4

because values sufficiently close to

2

are near

4

. d) The limit does not exist because

g(2)

is not provided in the table.

Intuitive explanation

This problem asks you to read a table and detect what value $g(x)$ is approaching from both sides as $x$ gets closer to 2—even though $g$ may not be defined at $x=2$ itself. Notice that from the left (1.9, 1.99, 1.999) the outputs cluster around one value, while from the right (2.001, 2.01, 2.1) they cluster around a different value. A two-sided limit exists only when both one-sided limits agree; here they don't, so the limit does not exist.

Intuitive explanation

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