Question

EXAMPLE 2 Prove the following limit. lim_{x ? 3} 3x - 7 = 2 SOLUTION 1. Preliminary analysis of the problem (guessing a value for ?). Let ? be a given positive number. We want to find a number ? such that if 0 < |x - 3| < ? then |(3x - 7) - 2| < ?. But |(3x - 7) - 2| = |3x - 9| = 3| |. Therefore, we want ? such that if 0 < |x - 3| < ? then 3| | < ? that is, if 0 < |x - 3| < ? then | | < ?/3. This suggests that we should choose ? = ?/3. 2. Proof (showing that ? works). Given ? > 0, choose ? = ?/3. If 0 < | | < ?, then |(3x - 7) - 2| = | | = 3| | < 3? = 3( ) = ?. Thus if 0 < |x - 3| < ? then |(3x - 7) - 2| < ?. Therefore, by the definition of a limit lim_{x ? 3} 3x - 7 = 2.

          EXAMPLE 2 Prove the following limit.
lim_{x ? 3} 3x - 7 = 2
SOLUTION 1. Preliminary analysis of the problem (guessing a value for ?). Let ? be a given positive number. We want to find a number ? such that
if 0 < |x - 3| < ? then |(3x - 7) - 2| < ?.
But |(3x - 7) - 2| = |3x - 9| = 3|    |. Therefore, we want ?
such that
if 0 < |x - 3| < ? then 3|    | < ?
that is, if 0 < |x - 3| < ? then |    | < ?/3.
This suggests that we should choose ? = ?/3.
2. Proof (showing that ? works). Given ? > 0, choose ? = ?/3. If
0 < |    | < ?, then
|(3x - 7) - 2| = |    |
= 3|    | < 3?
= 3(    )
= ?.
Thus
if 0 < |x - 3| < ? then |(3x - 7) - 2| < ?.
Therefore, by the definition of a limit
lim_{x ? 3} 3x - 7 = 2.

EXAMPLE 2 Prove the following limit.
limx ? 3 3x - 7 = 2
SOLUTION 1. Preliminary analysis of the problem (guessing a value for ?). Let ? be a given positive number. We want to find a number ? such that
if 0 < |x - 3| < ? then |(3x - 7) - 2| < ?.
But |(3x - 7) - 2| = |3x - 9| = 3| |. Therefore, we want ?
such that
if 0 < |x - 3| < ? then 3| | < ?
that is, if 0 < |x - 3| < ? then | | < ?/3.
This suggests that we should choose ? = ?/3.
2. Proof (showing that ? works). Given ? > 0, choose ? = ?/3. If
0 < | | < ?, then
|(3x - 7) - 2| = | |
= 3| | < 3?
= 3( )
= ?.
Thus
if 0 < |x - 3| < ? then |(3x - 7) - 2| < ?.
Therefore, by the definition of a limit
limx ? 3 3x - 7 = 2.

Added by Jacob S.

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