The value of the real number a for which the right hand limit limx→0+ (1−x)1/x−e−1xa is equal to a non-zero real number, is

It is given that limx→0+ (1−x)1/x−e−1xa exists and is a non-zero numberNow,limx→0+ (1−x)1/x−e−1xa=limx→0+ e1xlog⁡(1−x)−e−1xa=limx→0+ e−1+x2+x23+x34+⋯−e−1xa=limx→0+ e−1e−x2+x23+x34+…−1xa=limx→0+ e−1e−x2+x23+x34+…−1−x2+x23+x34+…×−1xax2+x23+x34+…=−e−1×1limx→0+ 12x1−a+13x2−a+14x3−a+…=−1e×12, if a=1.Hence, the limit is a non-zero number when a= 1.