18: After substituting the limit point into the given expression
you should get
![](gif8/ecb8ar6.gif)
Why is there 1+ in the logarithm? It is necessary to determine
whether for x close to but smaller than 5 the expression
(2x − 4)/(x + 1) is smaller than or
greater than 1. This cannot be done by playing with pluses and minuses,
since after substituting in one gets 6+/6+ which does
not help at all, all depends on a fine balance of numerator and denominator.
The easiest way is to solve some algebraic inequality, for instance one can
ask when is the ratio greater than 1. Turns out that such an inequality is
true for x > 5 (and for
x < −1),
which shows that as we approach 5 from the right, the ratio approaches 1
from larger numbers and it becomes 1+.
Hint on limit at
−∞
Hint on limit at
−1 from the left
Hint on limit at
2 from the right
Hint on limit at
5 from the left
Hint on limit at
∞
Answer