Problem 79 - Calculus for Biology and Medicine

Question

In Problems 79-90, use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{2}{n^{2}}\right) $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Expression

We need to find the limit as \( n \) approaches infinity of the expression \( \frac{1}{n} + \frac{2}{n^2} \). This expression is a sum of two fractions where the denominators involve powers of \( n \).

Step 2: Apply Limit Laws to Each Term

According to limit laws, the limit of a sum is equal to the sum of the limits. We'll handle each term separately: \( \lim_{n \to \infty} \frac{1}{n} \) and \( \lim_{n \to \infty} \frac{2}{n^2} \).

Step 3: Evaluate the First Term Limit

For the term \( \frac{1}{n} \), as \( n \) approaches infinity, \( \frac{1}{n} \) approaches 0 since the denominator becomes very large. Thus, \( \lim_{n \to \infty} \frac{1}{n} = 0 \).

Step 4: Evaluate the Second Term Limit

For the term \( \frac{2}{n^2} \), as \( n \) approaches infinity, \( \frac{2}{n^2} \) also approaches 0 because the denominator \( n^2 \) grows faster than \( n \). Thus, \( \lim_{n \to \infty} \frac{2}{n^2} = 0 \).

Step 5: Combine the Limits

Since both limits in the expression \( \frac{1}{n} + \frac{2}{n^2} \) are 0, using the limit sum rule, we add them: \( 0 + 0 = 0 \).

Step 6: State the Final Result

The limit of the entire expression as \( n \) approaches infinity is \( 0 \), so \( \lim_{n \to \infty} \left( \frac{1}{n} + \frac{2}{n^2} \right) = 0 \).

Key Concepts

Limit Laws

Limit laws in calculus provide essential rules for finding the limits of functions or sequences. Understanding these rules simplifies solving limit problems by breaking down complex expressions into manageable parts.
Here are some important limit laws you should know:

• **Sum Law**: The limit of the sum of two functions is equal to the sum of their individual limits. If we have functions \( f(n) \) and \( g(n) \), then:\[\lim_{n \to \infty} [f(n) + g(n)] = \lim_{n \to \infty} f(n) + \lim_{n \to \infty} g(n)\]
• **Constant Multiple Law**: If you multiply a function by a constant, the limit of the function is the constant times the limit of the function. For instance, \( c\cdot f(n) \) becomes:\[\lim_{n \to \infty} [c\cdot f(n)] = c \cdot \lim_{n \to \infty} f(n)\]

The limit laws are applied individually to each term in a sequence, using knowledge of how terms behave as \( n \to \infty \). When handling a sequence like \( \frac{1}{n} + \frac{2}{n^2} \), apply the Sum Law to distribute the limit operation across both terms separately, which simplifies the problem significantly.

Infinite Limit

An infinite limit describes the behavior of sequences or functions as variable values grow indefinitely. This concept is crucial when dealing with expressions where denominators approach zero or become extremely large, leading the entire term towards zero.
In the sequence \( \frac{1}{n} + \frac{2}{n^2} \), both terms demonstrate infinite limits.
Let's break it down:

• For \( \frac{1}{n} \), as \( n \) becomes very large, the fraction approaches zero. The more \( n \) increases, the smaller the value of the fraction, illustrating an infinite limit behavior that trends towards zero.
• Similarly, \( \frac{2}{n^2} \) behaves in the same manner. Since \( n^2 \) increases faster than \( n \), this term also dwindles to zero as \( n \) grows.

Understanding infinite limits helps predict the end behavior of terms involving growing denominators, leading to a better grasp of how sequences resolve as they extend.

Denominator Powers

The concept of denominator powers plays a significant role in understanding how fractions behave within limits. Higher powers in the denominator cause the overall value of a fraction to approach zero more quickly than lower powers.
Here's how it works:

• Consider the expressions \( \frac{1}{n} \) and \( \frac{2}{n^2} \). The term \( \frac{1}{n} \) reduces more slowly compared to \( \frac{2}{n^2} \) because \( n^2 \) grows faster than \( n \).
• In general, if you have \( \frac{1}{n^p} \), the larger the exponent \( p \), the faster the fraction trends towards zero. Consequently, sequences or functions with high power denominators often simplify to zero when \( n \to \infty \).

Understanding denominator powers helps simplify limit problems, showing why terms vanish as denominators grow infinitely large. This understanding is critical to predict results accurately in calculus limit problems.