The left hand limit of the function f(x)=|x−4|x−4, x≠40, x=4 at x = 4, is

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

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

Solution:

Related content