First slide

Introduction to limits

Question

The left hand limit of the function
$f (x) = \{\begin{cases} \frac{| x - 4 |}{x - 4}, x \neq 4 \\ 0, x = 4 \end{cases}$
at x = 4, is

Moderate

A

1
B

-1
C

0
D

non-existent

Solution

we have,
(LHL of $f (x)$ at $x = 4$ )
$\begin{aligned} = lim_{x \to 4^{-}} f (x) = lim_{h \to 0} f (4 - h) = lim_{h \to 0} \frac{| 4 - h - 4 |}{4 - h - 4} \\ = lim_{h \to 0} \frac{| - h |}{- h} = lim_{h \to 0 - h} \frac{h}{h} = lim_{h \to 0} - 1 = - 1 . \end{aligned}$
To evaluate RHL of $f (x) at x = a i.e, lim_{x \to a^{+}} f (x)$ we may use the following algorithm

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