Question

limit x tends to zero 16^x - 4^x / 4^x - 2^x​

Answer 1

As x→4 from the left, x<4⟹x−4<0⟹|x−4|=(4−x)=−(x−4).

Substitute −(x−4) for |x−4| into the original limit, and divide.

limx→4−x−4|x−4|=limx→4−(x−4)−(x−4)=−1.

after canceling the common factor in the numerator and denominator.

Similarly, as x→4 from the right, x>4⟹(x−4)>0⟹|x−4|=(x−4).

So

limx→4+x−4|x−4|=limx→4+(x−4)(x−4)=1.