Question

✬ Hey There!

Evaluate $\sf \: lim_{x \to 0} \: f(x)$ Where, $\sf \: f(x) = \begin{cases} \sf \dfrac{|x|}{x}, \: \: x \neq 0\\ \sf x \\ \sf 0, \: x = 0 \end{cases}$
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Answer 1

ANSWER⤵️

To find the limit at x=0 we should have

$\implies{lim. \: f(x ) \: = \: lim \: f(x) \: lim \: f (x)}$

x→0-. x→0+ x→0

$\implies{lim. f(x) = lim \frac{ |x| }{x} = \frac{ - x}{x} = -1}</p><p> </p><p>$

x→0- x→0-

since ∣x∣= −x for x < 0

$lim \: f(x) = lim \frac{ |x| }{x } = \frac{x}{x } = 1$

x→0+ x→0+

since ∣x∣= x for x > 0

Thus,

$\implies \pink {lim \: f(x) \: not \: equal \: to \: lim \: f(x) }$

x→0- x→0+

HENCE, lim f(x) does not exist.

x→0

$\blue{BE \: \: BRÃÎÑLY}$♡

✝ HOPE IT HELPS✝

Answer 2

SOLUTION

TO DETERMINE

$\displaystyle \sf{ \lim_{x \to 0} \: f(x)}$

Where

$\sf f(x) = \begin{cases} & \sf{ \: \: \: \displaystyle \sf{} \frac{ |x| }{x} \: \: \: \: when \: x \ne \: 0} \\ \\ & \sf{ \: \: \: \: \: 0 \: \: \: \: \: when \: x = 0} \end{cases}$

CONCEPT TO BE IMPLEMENTED

A function f(x) is said to have a limit L at x = a if

$\displaystyle \sf{ \lim_{x \to a + } \: f(x) =\lim_{x \to a - } \: f(x) = L}$

EVALUATION

Here the given function is

$\sf f(x) = \begin{cases} & \sf{ \: \: \: \displaystyle \sf{} \frac{ |x| }{x} \: \: \: \: when \: x \ne \: 0} \\ \\ & \sf{ \: \: \: \: \: 0 \: \: \: \: \: when \: x = 0} \end{cases}$

Now by the definition of modulus

$\sf |x | = \begin{cases} & \sf{ \: \: \: \: x \: \: \: \: when \: x > 0} \\ \\ & \sf{ \: - x \: \: \: \: \: when \: x \leqslant 0} \end{cases}$

Now

$\displaystyle \sf{ \lim_{x \to 0 + } \: f(x)}$

$\displaystyle \sf{ = \lim_{x \to 0 + } \: \frac{ |x| }{x} }$

$\displaystyle \sf{ = \lim_{x \to 0 + } \: \frac{ x }{x} }$

$= 1$

Again

$\displaystyle \sf{ \lim_{x \to 0 - } \: f(x) }$

$\displaystyle \sf{ = \lim_{x \to 0 - } \: \frac{ |x| }{x} }$

$\displaystyle \sf{ = \lim_{x \to 0 - } \: \frac{ - x}{x} }$

$= - 1$

$\displaystyle \sf{ \therefore \: \: \lim_{x \to 0 + } \: f(x) \ne \: \lim_{x \to0 - } \: f(x) }$

$\displaystyle \sf{ \therefore \: \: \lim_{x \to 0} \: f(x) \: \: does \: not \: exists \: }$

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