How to solve limits tending to -infinity with modulus?
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Assuming mod means |x|.
The function's value at x=0x=0 is not defined, and hence we try to take limits from left and right side.
Note that |x|=x,x≥0|x|=x,x≥0 and |x|=−x,x<0|x|=−x,x<0.
Therefore,
limx→0+|x|x=limx→0+xx=1limx→0+|x|x=limx→0+xx=1.
and
limx→0−|x|x=limx→0−−xx=−1limx→0−|x|x=limx→0−−xx=−1.
Hence, the left and right sided limits do not match and the function is not continuous atx=0x=0.
The function's value at x=0x=0 is not defined, and hence we try to take limits from left and right side.
Note that |x|=x,x≥0|x|=x,x≥0 and |x|=−x,x<0|x|=−x,x<0.
Therefore,
limx→0+|x|x=limx→0+xx=1limx→0+|x|x=limx→0+xx=1.
and
limx→0−|x|x=limx→0−−xx=−1limx→0−|x|x=limx→0−−xx=−1.
Hence, the left and right sided limits do not match and the function is not continuous atx=0x=0.
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