Problem 23 - Calculus: Early Transcendentals

Question

Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of \(f(a), \lim _{x \rightarrow a^{-}} f(x), \lim _{x \rightarrow a^{+}} f(x),\) and lim \(f(x)\) or state that they do not exist. $$f(x)=\frac{x^{2}-25}{x-5} ; a=5$$

Step-by-Step Solution

Answer

Answer: f(a) does not exist, the limit as x approaches 5 from the left and right is 10, and the overall limit is also 10.

Step 1: Simplify the function if possible

First, let's analyze the function's structure and see if we can simplify it. We are given \(f(x) = \frac{x^2 - 25}{x-5}\). Notice that the numerator is a difference of two squares, which we can simplify: $$(x^2 - 25) = (x+5)(x-5).$$ Thus, our function becomes: $$f(x) = \frac{(x+5)(x-5)}{x-5}.$$ When x is not equal to 5, we can cancel out the (x-5) term, which gives us: $$f(x) = x + 5.$$ So, the function is a simplified linear function, except at x=5, where f(x) is not defined.

Step 2: Plot the simplified function

Now, let's plot the simplified linear function \(f(x) = x+5\). This is a straight line with a slope of 1 and passing through the point (0, 5). Be sure to omit the point (5, 10) as our original function is not defined at x=5.

Step 3: Determine f(a), \(\lim _{x \rightarrow a^{-}} f(x)\), and \(\lim _{x \rightarrow a^{+}} f(x)\)

Based on the graph, we make conjectures about the values of f(a), the limit as x approaches a from the left, and the limit as x approaches a from the right. - f(a): As f(5) is not defined in our original function, f(a) does not exist. - \(\lim _{x \rightarrow a^{-}} f(x)\): As x approaches 5 from the left, the function acts just like the linear function, approaching a value of 10. So, we can conjecture that the limit as x approaches 5 from the left is 10. - \(\lim _{x \rightarrow a^{+}} f(x)\): Similarly, as x approaches 5 from the right, the function acts like the linear function, also approaching a value of 10. So, we can conjecture that the limit as x approaches 5 from the right is 10.

Step 4: Determine \(\lim_{x \rightarrow a} f(x)\) or state it doesn't exist

Now, let's determine if the overall limit as x approaches 'a' (5) exists or not. Since \(\lim _{x \rightarrow a^{-}} f(x) = 10\) and \(\lim _{x \rightarrow a^{+}} f(x) = 10\), both limits are the same, which means the overall limit exists and \(\lim_{x\rightarrow 5} f(x) = 10\). In summary, f(a) does not exist, and the limits of the function as x approaches 5 from the left, from the right, and overall are all equal to 10.

Key Concepts

Graphical Analysis

Graphical analysis is a valuable tool for understanding the behavior of functions visually. When faced with a function like \( f(x) = \frac{x^2 - 25}{x-5} \), sketching its graph can give insights into limits and continuity. Let’s break it down:

• Start by simplifying the function. In our case, the expression \( x^2 - 25 \) can be rewritten using the difference of squares, giving us \( (x+5)(x-5) \).
• Cancel out the common \( (x-5) \) term in the numerator and denominator, resulting in a simplified linear function \( f(x) = x + 5 \), but remember it's undefined at \( x = 5 \).
• Plot this line, keeping in mind the point \( (5, 10) \) should be omitted since the function is not defined there.

By observing the behavior around \( x = 5 \) on the graph, you can make accurate conjectures about the limits and their existence.

Difference of Squares

The difference of squares is a mathematical technique that simplifies expressions like \( x^2 - 25 \). It is based on the identity \( a^2 - b^2 = (a+b)(a-b) \), which can be very useful in calculus.

For the function \( f(x) = \frac{x^2 - 25}{x-5} \), it allows you to rewrite it as \( \frac{(x+5)(x-5)}{x-5} \). After canceling the \( (x-5) \) terms, you get \( f(x) = x + 5 \) for all \( x eq 5 \).

Here's how it helps:

• It transforms a complex expression into something much simpler and manageable.
• By doing so, you can more easily investigate limits and other properties of the function.

Remember, always check the domain of the original and simplified functions to understand where the function might be undefined.

Undefined Function

An undefined function means there are certain points where the function does not have a valid result. In the example \( f(x) = \frac{x^2 - 25}{x-5} \), the function is undefined at \( x = 5 \) because it causes division by zero.

This is crucial when determining limits:

• The original function doesn't exist at that point, so \( f(5) \) is undefined.
• However, you can still evaluate the behavior around that point using limits.

Since the limits from the left and right are both 10, you can conclude that though \( f(5) \) does not exist, \( \lim_{x \to 5} f(x) = 10 \). By carefully analyzing the simplified function, you can navigate around undefined points and understand the overall behavior.