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in Limits and Continuity by (49.1k points)
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Evaluate the limit: limx → 1 (xm - 1)/(xn - 1), m and n are integers.

2 Answers

+1 vote
by (38.6k points)
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Best answer

\(\lim\limits_{x \to 1}\frac{x^{m-1}}{x^n-1}(\frac00 type)\)

\(=\lim\limits_{x\to 1}\frac{mx^{m-1}}{nx^{n-1}}\) (By D.L.H. Rule)

\(=\frac{m(1)^{m-1}}{n(1)^{n-1}}=\frac mn\)

Alternative:

\(\lim\limits_{x\to 1}\frac{x^{m}-1}{x^n-1}\) = \(\lim\limits_{x\to 1}\cfrac{\frac{x^m-1}{x-1}}{\frac{x^n-1}{x-1}}\)

\(=\lim\limits_{x\to 1}\frac{x^{m-1}+x^{m-2}+....x^2+x+1}{x^{n+1}+x^{n-2}+....x^2+x+1}\)

\(=\frac{1^{m+1}+1^{m-2}+1^2+1+1(m\,terms)}{1^{n-1}+1^{n-2}+...+1^2+1+1 (n\,terms)}\)

\(=\frac{1+1+...+1+1+1(m\,terms)}{1+1+...+1+1+1(n\,terms)}\) 

\(=\frac mn\)

+2 votes
by (46.9k points)

= m(1)m - 1/n(1)n - 1 = m/n

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