Problem 44 - Contemporary Precalculus

Question

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 1} \tan \pi x$$

Step-by-Step Solution

Answer

Answer: The limit of the function \(f(x) = \tan(\pi x)\) as \(x\) approaches 1 is 0, and it is equal to the function's value at the given point.

Step 1: Graph the function

To start, let's graph the function \(f(x) = \tan(\pi x)\). This will give us an idea of how the function behaves around \(x=1\) and will aid us in determining whether the limit exists. You may use a graphing calculator or an online graphing tool like Desmos to plot the function.

Step 2: Observe the behavior of the function as x approaches 1

After plotting the function, we can observe its behavior as \(x\) approaches 1. You should notice that the function appears to approach a certain value as \(x\) gets closer to 1. This suggests that the limit may exist at this point.

Step 3: Find the limit of the function as x approaches 1

To find the limit of \(f(x)\) as \(x \rightarrow 1\), notice that we can simply evaluate \(f(1)\). Remember, the function is defined as \(f(x) = \tan(\pi x)\). So, we plug in \(x=1\) and evaluate the function: \(f(1) = \tan(\pi \times 1) = \tan(\pi) = 0\) This indicates that the limit of the function as \(x\) approaches 1 is 0. So, we can write: $$\lim _{x \rightarrow 1} \tan \pi x = 0$$

Step 4: Check if the limit equals the value of the function at the given point

Now, we have to verify if the limit found in Step 3 is equal to \(f(c)\) where \(c=1\). We know that \(f(1) = \tan(\pi \times 1) = \tan(\pi) = 0\). Since the limit of the function as \(x\) approaches 1 is also 0, we can conclude that the limit is indeed equal to the function's value at the given point: $$\lim _{x \rightarrow 1} \tan \pi x = f(1)$$ In conclusion, the limit of the function \(f(x) = \tan(\pi x)\) as \(x\) approaches 1 is 0, and this limit is equal to the function's value at \(x=1\).

Key Concepts

Graphical Analysis of Limits

Graphical analysis is a powerful method for understanding the behaviorof functions, especially when we're dealing with limits. By visualizingthe function, you can see how it behaves around a specific point,which helps in estimating limits intuitively.
When we talk about the function \(f(x) = \tan(\pi x)\), creating a graphcan provide crucial insight. Use a graphing tool, like a graphing calculatoror an online platform such as Desmos, to plot the function.As you plot, observe the path of the graph as \(x\) nears 1.
You'll notice that the function tends towards a constant value,indicating that the limit might exist. In our case, the graph showsthe function approaching 0. This insight primes us for deeper numericalor algebraic analysis, allowing us to confirm our findings formally.

Numerical Methods to Find Limits

Numerical methods offer another approach to finding limits, particularlywhen the visually-based approach of graphical analysis isn't enough.Using numbers, we can systematically evaluate the limit by checkingfunction values near the point of interest.
For \(f(x) = \tan(\pi x)\), evaluate the function at various near-1 values,such as 0.9, 0.95, 0.99, 1.01, 1.05, and 1.1.

• Calculate \(f(0.9)\), \(f(0.95)\), \(f(0.99)\) and so on.
• Observe as the results consistently trend towards 0.

This shows the limit as \(x\) approaches 1 appears to be 0.Although graphing is visual, numerical methods help verifyand provide confidence in our results. Combining these approachescan often make tricky problems easier to handle.

Understanding Trigonometric Functions

Trigonometric functions are fundamental in mathematics, dealingwith the relationships between angles and sides of triangles.The tangent function, \(\tan(x)\), is one of these and can be complexdue to its periodic nature.
When you encounter \(\tan(\pi x)\), the function becomes periodic withasymptotic behavior at some points.
For \(x=1\), evaluating \(\tan(\pi x)\) yields \(\tan(\pi)\),which equals 0. This is an important result: knowing howthe function behaves at specific points helps us find limits anddetermine whether these match the function's original value.
Understanding these periodic patterns, alongside analyzing behaviorusing graphs and numbers, helps decode the function's operationacross different inputs. It's fascinating to see how trigonometric functionslike tangent weave through mathematics with unique challengesand powerful tools.