What Went Wrong?

Calculus Level 2

Is the following sequence of steps valid? If no, which step is wrong?

Step 1: Consider the following limit:

\[ L = \lim_{n \rightarrow \infty} \frac { 1 + 2 + 3 + \cdots + n } { n^2 } \]

Step 2: Verify that it is of the indeterminate form \( \dfrac { \infty}{\infty} \).

  • Numerator tends to infinity
  • Denominator tends to infinity

Step 3: Apply L'hôpital's rule to get

\[ L = \lim_{n \rightarrow \infty} \frac{ 1 } { 2n } = 0 \; . \]

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