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Evaluate ("lim")(x->0)f(x),\ w h e r e\ ...

Evaluate `("lim")_(x->0)f(x),\ w h e r e\ f(x)={(|x|)/x ,\ x!=0 , 0,\ x=0`

Text Solution

Verified by Experts

To find the limit at x=0 we should have
`lim_(x->0^-)f(x)=lim_(x->0^+)f(x)lim_(x->0)f(x)`
`lim_(x->0^-)f(x)=lim_(x->0^-)(absx)/x=-x/x=-1` since `absx`=`-x , x<0`
`lim_(x->0^+) f(x)=lim_(x->0^+)(absx)/x=x/x=1` since `absx`=`x, x>0`
Thus `lim_(x->0^-)f(x)!=lim_(x->0^+)f(x)`
Hence, `lim_(x->0)f(x)` does not exist.
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