Problem 18 - Thomas Calculus

Question

Find the limits in Exercises \(11-22\) $$\lim _{y \rightarrow 2} \frac{y+2}{y^{2}+5 y+6}$$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{5}\).

Step 1: Evaluate the Expression Directly

First, we'll attempt to evaluate the limit by directly substituting the value of the variable into the expression. Substitute \(y = 2\) into \(\frac{y+2}{y^2+5y+6}\):\[\lim_{y \to 2} \frac{y+2}{y^2+5y+6} = \frac{2+2}{2^2 + 5 \cdot 2 + 6}\]Simplifying the term:\[= \frac{4}{4 + 10 + 6} = \frac{4}{20}\]

Step 2: Simplify the Result

Simplify the fraction from the direct substitution:\[\frac{4}{20} = \frac{1}{5}\]Since substituting \(y = 2\) did not result in an undefined form (like 0/0), we've found the limit without needing to further manipulate the expression.

Key Concepts

Direct Substitution

Direct substitution is one of the simplest methods used to find the limit of a function as a variable approaches a specific value. In mathematics, we often use direct substitution when the expression does not result in an indeterminate form. When you simply plug in the approaching value into the function, you can directly calculate the result if it yields a valid number.

In the exercise provided, we demonstrate direct substitution with the expression \( \lim_{y \to 2} \frac{y+2}{y^2+5y+6} \). By substituting \( y = 2 \) directly into the expression:

• We substitute: \( \frac{2 + 2}{2^2 + 5 \times 2 + 6} \)
• The expression becomes: \( \frac{4}{4 + 10 + 6} = \frac{4}{20} \)

Clearly, the value \( \frac{4}{20} \) is not undefined or indeterminate, allowing us to directly find the limit without any further simplification or transformation of the expression.

Direct substitution is a reliable method for basic limit evaluations when the expression resolves cleanly to a numerical value. If substituting the value results in 0/0 or another indeterminate form, we must explore alternate methods like simplifying the expression or using limit properties.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its lowest terms so it is easier to work with. When dealing with expressions that include limits, this step can be crucial to identify and eliminate potential indeterminate forms.

In our exercise, after performing direct substitution, we achieved the fraction \( \frac{4}{20} \). This can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which in this case is 4. By simplifying, we perform the following calculation:

• \( \frac{4}{20} = \frac{4 \div 4}{20 \div 4} \)
• This results in \( \frac{1}{5} \)

Thus, the fraction is reduced to its simplest form.

Simplifying fractions not only presents the solution more clearly but also confirms when the substitution doesn't result in an indeterminate expression. This practice ensures that unnecessary complexity doesn't cloud the solution process.

Mathematical Limits

The concept of a limit is central to calculus and understanding how functions behave as they approach a certain point. A limit formally describes the value that a function approaches as the input approaches some value. It is essential to precisely define the behavior of functions that may not be intuitively clear at certain points.

For the given exercise, the limit being evaluated is \( \lim_{y \to 2} \frac{y+2}{y^2+5y+6} \). The aim is to understand how the function behaves as \( y \) nears 2. As established through the process of direct substitution and simplifying fractions, our function approaches the value \( \frac{1}{5} \) as \( y \to 2 \).

Limits serve several purposes:

• Enable us to handle expressions with instabilities or discontinuities at given points.
• Help in defining derivatives and integrals, backing up essential calculus operations.
• Assist in predicting the behaviour of functions near specific values, providing insights into graph behavior.

By mastering limits, one gains deeper insight into both basic and complex functions, allowing not just evaluation, but also enabling analytical prowess in calculus and beyond.