Evaluate each of the following, if it exists. (If an answer does not exist, enter DNE.)

Question

g(x) =	x	if x < 1
6		if x = 1
2 − x2		if 1 < x ≤ 2
x − 1		if x > 2

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(a) Evaluate each of the following, if it exists. (If an answer does not exist, enter DNE.)

(i)    lim x→1⁻ g(x)

(ii)    lim x→1⁺ g(x)

(iii)    g(1)

(iv)    lim x→2⁻ g(x)

(v)    lim x→2⁺ g(x)

(vi)    lim x→2 g(x)

Answer 1

(i) This will be 1. As you approach x = 1 from the left g(x) gets arbitrarily close to 1. (The limit is never actually attained but that is not necessary)

(ii) This will also be 1. As you approach x from the right 2-x² gets arbitrarily close to 1. (again the limit is not actually attained).

(iii) g(1) = 6, read from definition

(iv) 2 - 2² = -2

(v) 2 - 1 = 1

(vi) DNE. For this to exist left hand limit (iv) and right hand limit (v) must be equal which they are not.

Mike