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