First slide

Introduction to limits

Question

If $f (x) = |x - 1| - [x],$ where [x] is the greatest integer less than or equal to x, then

Moderate

A

$f (1 + 0) = - 1, f (1 - 0) = 0$
B

$f (1 + 0) = 0 = f (1 - 0)$
C

$\lim_{x \to 1} f (x) exists$
D

$\lim_{x \to 1} f (x) does exists$

Solution

$\begin{array}{l} f (1 + 0) = \lim_{h \to 0} (|1 + h - 1| - [1 + h]) = \lim_{h \to 0} (x - 1) = - 1 \\ f (1 - 0) = \lim_{h \to 0} (|1 - h - 1| - [1 - h]) = \lim_{h \to 0} (h - 0) = 0 \end{array}$

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