Question

Find the value of f(2) so that $f\left(x\right)=\frac{2^{x+8}-1024}{4^x-16}$ is continuous.

Answer 1

$f\left(2\right)=\underset{x\to 2}{lim}\frac{2^{x+8}-1024}{4^x-16}=\underset{x\to 2}{lim}\frac{2^8(2^x-4)}{(2^x{)}^2-4^2}$

$f\left(2\right)=\underset{x\to 2}{lim}\frac{2^8(2^x-4)}{(2^x+4)(2^x-4)}=\underset{x\to 2}{lim}\frac{2^8}{(2^x+4)}$

$\Rightarrow f\left(2\right)=\frac{2^8}{8}=32$