Question

Please help evaluate:

lim b → s [( b ^4− s^4 )/( b ^5− s^5 )]

Answer 1

Let s = 0
\[lim_{b→0}\frac{b^4}{b^5}=\lim_{b→0} \frac{1}{b}=±∞
\]
Let s = 1
\[lim_{b \rightarrow 1}\frac{b^4−1}{b^5−1}=\lim_{b\rightarrow 1}\frac{\frac{d(b^4−1)}{db}}{\frac{d(b^5−1)}{db}}=\lim_{b\rightarrow1}\frac{4b^3}{5b^4}=\lim_{b\rightarrow1}\frac{4}{5b} = \frac{4}{5}\]

Let s = any real number
\[lim_{b \rightarrow s}\frac{b^4−c}{b^5−c'}=\lim_{b\rightarrow s}\frac{\frac{d(b^4−c)}{db}}{\frac{d(b^5−c')}{db}}=\lim_{b\rightarrow s}\frac{4b^3}{5b^4}=\lim_{b\rightarrow s}\frac{4}{5b} = \frac{4}{5s}\]

Where c and c' are constants
So we can ignore the irregularity at b->s when s is 0 and just say the limit is 4/5s everywhere.