Problem 29 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}} $$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{6} \).

Step 1: Understanding the Limit Expression

We need to evaluate \( \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}} \). Look for any indeterminate forms as \( x \to 0 \). Notice that both the numerator and denominator tend to zero, indicating a \( \frac{0}{0} \) form.

Step 2: Simplifying the Expression

To simplify, multiply the numerator and the denominator by the conjugate of the numerator: \( \sqrt{x^{2}+9} + 3 \). This gives\[\frac{(\sqrt{x^{2}+9}-3)(\sqrt{x^{2}+9}+3)}{x^{2}(\sqrt{x^{2}+9}+3)}\].

Step 3: Applying the Conjugate Product

Apply the formula for the difference of squares: \((a - b)(a + b) = a^2 - b^2\). We get\[\frac{x^2}{x^{2}(\sqrt{x^{2}+9}+3)}\].

Step 4: Canceling Terms

Now, cancel the \(x^2\) terms from the numerator and denominator, resulting in\[\frac{1}{\sqrt{x^2 + 9} + 3}\].

Step 5: Evaluating the Limit

Substitute \( x = 0 \) into the remaining expression:\[\lim_{x \to 0} \frac{1}{\sqrt{0^2 + 9} + 3} = \frac{1}{3 + 3} = \frac{1}{6}\].

Step 6: Concluding the Solution

After evaluating the expression in the steps above, we determine the value of the limit is \( \frac{1}{6} \). This can also be confirmed using a table of values that approach \( x = 0 \) from both sides.

Key Concepts

Limit problems

Limit problems often arise in calculus when we want to find the value that a function approaches as the input approaches a certain point. These problems help determine the behavior of functions at points where they might not be explicitly defined. In our exercise, we deal with the limit as the variable approaches zero for the expression \( \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}} \). The goal is to find what value this expression gets infinitely close to, when \( x \) is very near to zero but not exactly zero.When solving limit problems, one might encounter difficulties due to forms like \( \frac{0}{0} \) or \( \frac{\infty}{ny} \), known as indeterminate forms. These expressions require special techniques to resolve, as direct substitution does not give a clear answer.

Conjugate method

The conjugate method is a powerful algebraic tool used to simplify expressions, especially when dealing with square roots. It involves multiplying the expression by a conjugate to eliminate radicals, making it easier to handle. This method is particularly useful for solving limit problems that feature indeterminate forms.In our exercise, the conjugate of the numerator \( \sqrt{x^{2}+9}-3 \) is \( \sqrt{x^{2}+9}+3 \). By multiplying both the numerator and the denominator by this conjugate, the expression simplifies significantly:

• It transforms \( (\sqrt{x^{2}+9}-3)(\sqrt{x^{2}+9}+3) \) into a difference of squares, which simplifies the calculation.

As a result, this process reduces the limit expression into a form that no longer contains the indeterminate \( \frac{0}{0} \) and can be easily evaluated.

Indeterminate forms

Indeterminate forms in calculus refer to expressions that do not directly lead to a specific limit. Common indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{ny} \), \( 0 \times \infty \), among others. These forms arise when trying to evaluate limits directly without additional algebraic manipulation.In our problem, as \( x \) approaches zero, the expression \( \frac{\sqrt{x^{2}+9}-3}{x^{2}} \) results in a \( \frac{0}{0} \) form. Evaluating such forms requires maneuvers like factoring, rationalizing using the conjugate, or applying L'Hpital's Rule in more advanced scenarios. These techniques help convert indeterminate forms into determinate ones, making limit evaluation possible.

Difference of squares

The difference of squares is a specific algebraic identity that greatly simplifies expressions in calculus and algebra. The identity states that \( (a - b)(a + b) = a^2 - b^2 \). By recognizing situations where this can be applied, complex expressions often become manageable.In the context of our problem, after multiplying by the conjugate, we employ this identity to handle the expression \( (\sqrt{x^{2}+9}-3)(\sqrt{x^{2}+9}+3) \):

• This becomes \( x^2 + 9 - 9 \), simplifying to \( x^2 \).
• This results in an easier-to-solve limit expression \( \frac{x^2}{x^2(\sqrt{x^{2}+9}+3)} \), where the \( x^2 \) terms conveniently cancel out.

By using the difference of squares, the original expression's complexity is reduced, making it straightforward to determine limits.