Limited x-0(1+nx)1/x=?

1 Answer
Feb 27, 2018

#e^n#

Explanation:

Plugging in value of #x#, this is #1^oo# form.
As any number #a# can be written as #e^lna#

#lim_(xto0) e^(ln(1+nx)^(1/x)#

#lim_(xto0) e^((1/x)(ln(1+nx))#

Multiplying and dividing by #n# to use standard limit :
#lim_(xto0) ln(1+x)/x =1#

#lim_(xto0) e^((1/(nx))(ln(1+nx)n)#

#lim_(xto0) e^(ln(1+nx)/(nx)n#

Limit can be taken inside the exponential.

#e^(lim_(xto0)(ln(1+nx)/(nx)n)#

The limit is just #1#

#e^n#