\lim_{n \to \infty} \left( \frac{1}{n} - \frac{1}{5n^5} \right) = \lim_{n \to \infty} \frac{1}{n} - \lim_{n \to \infty} \frac{1}{5n^5}

["Understanding Limits: Proving That (\lim_{n \ o \infty} \left( \frac{1}{n} - \frac{1}{5n^5} \right) = 0)", "When analyzing limits in calculus, one common challenge is evaluating expressions involving sequences or functions as ( n ) approaches infinity. A helpful mathematical principle allows us to simplify complex expressions involving limits by breaking them into individual components — and this leads us naturally to proving:", "[\n\lim_{n \ o \infty} \left( \frac{1}{n} - \frac{1}{5n^5} \right) = \lim_{n \ o \infty} \frac{1}{n} - \lim_{n \ o \infty} \frac{1}{5n^5}\n]", "### What Does This Mean?", "The expression\n[\n\frac{1}{n} - \frac{1}{5n^5}\n]\ninvolves two terms that both dependence critically on ( n ) as it grows without bound. Instead of evaluating the entire expression directly — which requires understanding the behavior of each term in the limit — we can use a fundamental limit law:", "[\n\lim_{n \ o \infty} \left( a - b \right) = \lim_{n \ o \infty} a - \lim_{n \ o \infty} b\n]\nprovided both ( \lim_{n \ o \infty} a ) and ( \lim_{n \ o \infty} b ) exist. This expansion is allowed because limits behave well over subtraction.", "### Evaluating Each Term Independently", "We now evaluate each part separately:", "1. First term:\n[\n\lim_{n \ o \infty} \frac{1}{n}\n]\nAs ( n \ o \infty ), ( \frac{1}{n} \ o 0 ). This is a basic result: dividing 1 by increasingly large ( n ) yields values that shrink toward zero.", "2. Second term:\n[\n\lim_{n \ o \infty} \frac{1}{5n^5}\n]\nSince ( n^5 \ o \infty ) as ( n \ o \infty ), the entire fraction ( \frac{1}{5n^5} ) approaches 0. Constants (like ( \frac{1}{5} )) don’t affect the limit:\n[\n\lim_{n \ o \infty} \frac{1}{5n^5} = \frac{1}{5} \cdot \lim_{n \ o \infty} \frac{1}{n^5} = 0\n]", "### Bringing It All Together", "Putting both results together:", "[\n\lim_{n \ o \infty} \left( \frac{1}{n} - \frac{1}{5n^5} \right) = \lim_{n \ o \infty} \frac{1}{n} - \lim_{n \ o \infty} \frac{1}{5n^5} = 0 - 0 = 0\n]", "Thus, the limit converges to 0, confirming the equality directly derived from the limit law.", "### Why This Approach Helps", "Breaking the expression into limits before subtracting ensures correctness by leveraging known limit behaviors for rational functions and powers of ( n ). It avoids misapplying operations that might not preserve limit properties — especially important with infinite sequences, where intuitive arithmetic on infinite terms can lead to errors.", "### In Summary", "To evaluate\n[\n\lim_{n \ o \infty} \left( \frac{1}{n} - \frac{1}{5n^5} \right)\n]\nanalyzing the two components independently is both valid and efficient. Since both ( \frac{1}{n} ) and ( \frac{1}{n^5} ) approach 0 as ( n \ o \infty ), their difference also approaches 0:", "[\n\boxed{ \lim_{n \ o \infty} \left( \frac{1}{n} - \frac{1}{5n^5} \right) = 0 }\n]", "This illustrates a foundational technique in calculus: limits of differences equal the difference of limits, offered powerful and reliable when functions behave predictably at infinity.", "---", "Keywords: limit as n approaches infinity, (\lim_{n \ o \infty} \left( \frac{1}{n} - \frac{1}{5n^5} \right)), rational limits, calculus limit laws, (\frac{1}{n} \ o 0), (\frac{1}{n^5} \ o 0), mathematical proof, sequence limits."]