Show that lim x - 0x-x- does not exist-

Let f(x) =x/|x|. then x → 0^+limf(x) h → 0lim f(0+h)=h→ 0 lim f(h) =h→ 0 limh/|h|=h→ 0 limh/h=1. x → 0^ limf(x) h → 0lim f(0 h)=h→ 0 lim f(h) = h→ 0 limh/| h|=h ...

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Standard XII

Mathematics

Limit

Show that lim...

Question

Show that $l i m x \to 0 \frac{x}{| x |}$ does not exist.

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Solution

Let $f (x) = \frac{x}{| x |} .$ then

$l i m x \to 0^{+} f (x) l i m h \to 0 f (0 + h) = l i m h \to 0 f (h) = l i m h \to 0 \frac{h}{| h |} = l i m h \to 0 \frac{h}{h} = 1.$

$l i m x \to 0^{-} f (x) l i m h \to 0 f (0 - h) = l i m h \to 0 f (h) = l i m - h \to 0 \frac{h}{| - h |} = l i m h \to 0 \frac{- h}{h} = - 1.$

$∴ l i m x \to 0^{+} f (x) \neq l i m x \to 0^{-} f (x) .$

Hence, $l i m x \to 0^{-} f (x)$ does not exist.

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Show that limx→0x|x| does not exist.

Let f(x)=x|x|. then limx→0+f(x)limh→0f(0+h)=limh→0f(h)=limh→0h|h|=limh→0hh=1. limx→0−f(x)limh→0f(0−h)=limh→0f(h)=lim−h→0h|−h|=limh→0−hh=−1. ∴limx→0+f(x)≠limx→0−f(x). Hence, limx→0−f(x) does not exist.

Show that $l i m x \to 0 \frac{x}{| x |}$ does not exist.