\( \lim _ { x \rightarrow r } \frac { x ^ { r } - 1 } { x - r } \)
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To understand this limit better, we can view it through the lens of calculus and the concept of derivatives! This limit expression is actually akin to finding the derivative of the function \( f(x) = x^r \) at the point \( x = r \). By applying the definition of the derivative, we can evaluate this limit as \( r \) gets closer to \( x \). An important note is to recognize that if we directly substitute \( x = r \) in the expression, we encounter an indeterminate form \( \frac{0}{0} \). So what can we do? A common technique is l'Hôpital's Rule, which allows us to differentiate the numerator and denominator to resolve this. Using l'Hôpital's Rule, we differentiate the top and the bottom: 1. The derivative of the numerator \( x^r - 1 \) is \( r x^{r - 1} \). 2. The derivative of the denominator \( x - r \) is \( 1 \). Now we take the limit again: \[ \lim_{x \to r} \frac{r x^{r - 1}}{1} = r r^{r - 1} = r^r. \] So the limit evaluates neatly to \( r^r \). Isn't calculus fun?