Problem 87 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(83-88\) , find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\) $$ f(x)=\frac{1}{x+3} $$

Step-by-Step Solution

Answer

Therefore, the limit is \(- \frac{1}{{(x+3)^2}}\).

Step 1: Substitute the function into the limit formula

Substitute \(f(x) = \frac{1}{{x+3}}\) and \(f(x+{\Delta x}) = \frac{1}{{x+{\Delta x}+3}}\) into the limit formula to get \(\lim_{{\Delta x} \rightarrow 0} \frac{{\frac{1}{{x+{\Delta x}+3}} - \frac{1}{{x+3}}}} {\Delta x}\).

Step 2: Simplifying the Complex Fraction

Since we are dealing with a complex fraction (a fraction where the numerator and the denominator are also fractions), it can be simplified by multiplying the numerator and denominator by \(x+{\Delta x}+3)(x+3)\). This results in \(\lim_{{\Delta x} \rightarrow 0} \frac{{(x + 3) - (x + {\Delta x} + 3)}} {{\Delta x}(x+3)(x+{\Delta x}+3)} = \lim_{{\Delta x} \rightarrow 0} \frac{{-{\Delta x}}}{\Delta x(x+3)(x+{\Delta x}+3)}\)

Step 3: Cancelling Delta x and Taking the Limit

Cancelling \({\Delta x}\) from numerator and denominator, we get: \(\lim_{\Delta x \rightarrow 0} \frac{-1}{(x+3)(x+{\Delta x}+3)}\). Now, take the limit as \({\Delta x}\) approaches 0: \(- \frac{1}{(x+3)^2}\)

Key Concepts

Limit Formula

Understanding how to find limits in calculus is crucial, and the limit formula is at its core. The limit formula can be defined as \( \[ \lim_{{x \rightarrow a}} f(x) \] \) and is used to determine the value that a function \( f(x) \) approaches as \( x \) approaches a certain point \( a \). This concept is especially important when \( x \) is nearing a value where \( f(x) \) is not clearly defined or has an undefined form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

In the given exercise, the limit formula is applied to a function as \( \Delta x \) approaches zero. This calculation is essential in finding the derivative of a function at a point, which represents the rate of change of the function at that particular point. Simplifying the limit expression before taking the limit helps in finding the solution more easily, which is often done through algebraic manipulation such as factoring, expanding, or simplifying complex fractions.

Complex Fraction

A complex fraction is essentially a fraction where the numerator, the denominator, or both are also fractions. They often appear intimidating but can be simplified using common algebraic techniques. One standard method to simplify a complex fraction is to multiply the numerator and denominator by the Least Common Denominator (LCD) of all the fractions involved.

In the given step-by-step solution, the original complex fraction is \( \frac{{\frac{1}{{x+{\Delta x}+3}} - \frac{1}{{x+3}}}} {\Delta x} \). To simplify it, multiplying both the numerator and the denominator by \( (x+{\Delta x}+3)(x+3) \) eliminates the smaller fractions, leaving terms with \( \Delta x \) that can then be cancelled out. This is a pivotal step before applying the limit, ensuring that the function is easier to evaluate as \( \Delta x \) approaches zero.

L'Hôpital's Rule

When direct substitution in finding a limit leads to an indeterminate form, L'Hôpital's Rule comes to the rescue. This rule states that under certain conditions, if you have a limit in the form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) after direct substitution, you can take the derivatives of the numerator and the denominator separately and then find the limit again. L'Hôpital's Rule often simplifies the calculation of limits significantly.

In situations, however, where L'Hôpital's Rule is not applied, as in the exercise in question, it's due to the fact that once the complex fraction is simplified, we no longer have an indeterminate form. After simplification and cancellation, the limit can be directly calculated without further complications. L'Hôpital's Rule is a handy tool in a mathematician's toolkit but recognizing when to apply it is as crucial as understanding how to apply it.