Limit problem 3 Nov

Question

!!Lim_(x->0){ (e^(1/x) - 1)/(e^(1/x) + 1) }!!

!!a)" "0!!

!!b)" "1!!

!!c)-1!!

!!d)!!Not exist

Solution

We can solve this problem checking for left hand and right hand limit.

For right hand limit, we can check for !!(0+h)!!

!!= Lim_(h->0){ (e^(1/h) - 1)/(e^(1/h) + 1) }!!

!!= Lim_(h->0) (e^(1/h)(1 - 1/e^(1/h)))/(e^(1/h)(1 + 1/e^(1/h))!!

!!= Lim_(h->0) (1 - 1/e^(1/h))/(1 + 1/e^(1/h))!!

Now putting !!h=0!!

!!= (1 - 1/e^(1/0))/(1 + 1/e^(1/0))!!

!!= (1 - 0)/(1 + 0)!!

!!= 1!!

For left hand limit, we can check for !!(0-h)!!

!!= Lim_(h->0){ (e^(1/-h) - 1)/(e^(1/-h) + 1) }!!

!!= Lim_(h->0){ (1/e^(1/-h) - 1)/(1/e^(1/-h) + 1) }!!

Now putting !!h=0!!

!!= (1/e^(1/0) - 1)/(1/e^(1/0) + 1)!!

!!= (0 - 1)/(0 + 1)!!

!!= -1!!

Since right hand and left hand limit are not equal, hence limit of !!Lim_(x->0){ (e^(1/x) - 1)/(e^(1/x) + 1) }!! does not exist.

Get it on Google Play