Question

How do you find the limit `lim (4^(1+x)-4^(1-x))/(2^(1+x)-2^(1-x))` as `x->0`?

Answer 1

We have that

#lim (4^(1+x)-4^(1-x))/(2^(1+x)-2^(1-x))= lim_(x->0) ((2^(1+x)+2^(1-x))*(2^(1+x)-2^(1-x)))/(2^(1+x)-2^(1-x))= lim_(x->0) ((2^(1+x)+2^(1-x))*(cancel(2^(1+x)-2^(1-x))))/(cancel(2^(1+x)-2^(1-x)))= lim_(x->0) (2^(1+x)+2^(1-x))=4#

We use the identity `a^2-b^2=(a+b)*(a-b)`

where `a=2^(1+x)` and `b=2^(1-x)`