Evaluating Limits of Sequences

Harnessing the Power of Limit Laws

You likely spent a good deal of time in your calculus course learning how to compute limits of functions like ${\lim }_{x\to \infty }(f(x)+g(x))$ by considering the limits of $f(x)$ and $g(x)$ separately. The good news is that these same rules apply directly to sequences.

Let's formalize this by looking at the Algebraic Limit Laws for sequences.

As you've just read, these laws are a direct parallel to what you already know. They allow us to deconstruct a complicated sequence into simpler components whose limits are known. For instance, knowing that ${\lim }_{n\to \infty }\frac{1}{n^k}=0$ for any positive integer $k$ becomes a fundamental building block.

Let's see this in action with a common type of sequence: one defined by a ratio of polynomials. The technique should be very familiar.

Limits of Composite Sequences

The limit laws can be extended further. If a sequence $a_n$ converges to a limit $L$, and we apply a function $f$ that is continuous at $L$, then the new sequence $f(a_n)$ will converge to $f(L)$.

This is a very useful result. For example, to find the limit of the sequence $b_n=\cos \left(\frac{\pi }{n}\right)$, we can first evaluate the "inner" sequence, $a_n=\frac{\pi }{n}$. We know ${\lim }_{n\to \infty }{a}_n=0$. Since the cosine function is continuous everywhere, and specifically at 0, we can conclude:

L'Hôpital's Rule for Indeterminate Forms

What about limits that result in indeterminate forms like $\frac{\infty }{\infty }$ or $\frac{0}{0}$? Your calculus instincts are correct: L'Hôpital's rule is the tool for the job. However, there is a subtle but important detail to consider. L'Hôpital's rule requires differentiation, but a sequence $a_n$, being defined only on the integers, is not a differentiable function. How do we resolve this?

This is where the theorem from our last lesson becomes critical. We find a differentiable function $f(x)$ that agrees with the sequence at integer values (i.e., $f(n)=a_n$), and then apply L'Hôpital's rule to that function.

The following video explains this conceptual point with great clarity.

With this rigorous foundation, let's apply the method to some concrete examples.

The technique of using logarithms is powerful and worth seeing in action. The following video walks through several more problems, including some classic examples that require clever algebraic manipulation before L'Hôpital's rule can be applied.

Try It Yourself

Now, apply these techniques to determine if the following sequences converge or diverge. If they converge, find the limit.

1. $a_n=\frac{8n^4+2n^2-1}{2n^4-5n^3}$
2. $b_n=\frac{\ln (n)}{n^2}$
3. $c_n=n^{1/n}$

Solutions:

1. Sequence $a_n$: This is a rational sequence. We divide the numerator and denominator by the highest power, $n^4$:
   ${\lim }_{n\to \infty }\frac{8+2/n^2-1/n^4}{2-5/n}=\frac{8+0-0}{2-0}=4$
   The sequence converges to 4.

2. Sequence $b_n$: This limit has the indeterminate form $\frac{\infty }{\infty }$. We consider the function $f(x)=\frac{\ln (x)}{x^2}$ and apply L'Hôpital's rule:
   ${\lim }_{x\to \infty }\frac{\ln (x)}{x^2}\stackrel{L^{\prime }H}{=}{\lim }_{x\to \infty }\frac{1/x}{2x}={\lim }_{x\to \infty }\frac{1}{2x^2}=0$
   The sequence converges to 0.

3. Sequence $c_n$: This has the indeterminate form ${\infty }^0$. We use the logarithmic trick. Let $L={\lim }_{n\to \infty }{n}^{1/n}$.
   $\ln (L)={\lim }_{n\to \infty }\ln (n^{1/n})={\lim }_{n\to \infty }\frac{1}{n}\ln (n)={\lim }_{n\to \infty }\frac{\ln (n)}{n}$
   This is now the form $\frac{\infty }{\infty }$, so we apply L'Hôpital's rule:
   $\ln (L)\stackrel{L^{\prime }H}{=}{\lim }_{n\to \infty }\frac{1/n}{1}=0$
   Since $\ln (L)=0$, we have $L=e^0=1$. The sequence converges to 1.

Conclusion

In this lesson, we significantly expanded our toolkit for analyzing sequences. By leveraging the connection to functions, we've shown how familiar rules from your prior calculus experience can be applied with precision to determine the limits of sequences.

The key takeaways are:

• The Algebraic Limit Laws (for sums, products, quotients, etc.) for sequences are identical to those for functions, allowing you to compute limits of complex sequences by breaking them into simpler parts.
• For indeterminate forms ($\frac{\infty }{\infty }$, $\frac{0}{0}$, $0^0$, etc.), L'Hôpital's rule is a primary tool. The rigorous approach involves finding a corresponding continuous function $f(x)$ and applying the rule to it.
• For indeterminate powers, the technique of taking the natural logarithm is essential to transform the expression into a quotient suitable for L'Hôpital's rule.

While these methods are incredibly powerful, they don't cover every scenario. For instance, how would you handle a sequence like $a_n=\frac{(-1{)}^n}{n}$ or $b_n=\frac{\cos (n)}{n^2}$? The numerators oscillate, so L'Hôpital's rule isn't directly applicable. In our next lesson, we will introduce the Squeeze Theorem for Sequences, a clever tool designed to handle exactly these kinds of cases.