Question

lim x → 0 ( x + 1 ) 5 − 1 x

Answer 1

5

Answer 2

The limit is of the $\frac{0}{0}$ indeterminate form. By substituting $y=x+1$, the limit transforms into $\lim_{y \to 1} \frac{y^5 - 1^5}{y-1}$, which is a standard limit form $\lim_{y \to a} \frac{y^n - a^n}{y-a} = n a^{n-1}$. Applying this formula with $a=1$ and $n=5$ gives $5 \cdot 1^{5-1} = 5$. Alternatively, using L'Hopital's rule, differentiate the numerator and denominator to get $\lim_{x \to 0} \frac{5(x+1)^4}{1}$, which evaluates to $5(0+1)^4 = 5$.