Question

Use analytical methods and/or a graphing utility en identify the vertical asymptotes (if any) of the following functions. $$f(x)=\frac{x^{2}-3 x+2}{x^{10}-x^{9}}$$

Short answer

Answer: The vertical asymptote of the function is x = 0.

Step-by-step solution

Factorize the numerator and the denominator

We start by factorizing the numerator and the denominator of the given function \(f(x)=\frac{x^{2}-3x+2}{x^{10}-x^{9}}.\) First, we factorize the numerator: \(x^2 - 3x + 2 = (x - 1)(x - 2)\) Then, we factorize the denominator: \(x^{10} - x^{9} = x^9(x - 1)\)

Simplify the function

Now we simplify the function by canceling the common factors in the numerator and the denominator: $$f(x) = \frac{(x-1)(x-2)}{x^9(x-1)} = \begin{cases} \frac{x-2}{x^9} & x\neq 1 \\ \text{undefined} & x=1 \end{cases}$$

Find the vertical asymptotes

We find the vertical asymptotes by determining the real values of x for which the denominator is zero, but the numerator is not zero. In the simplified function, the denominator equals zero when \(x = 0\). The numerator does not equal zero at this point, so \(x = 0\) is a vertical asymptote. The function is also undefined when \(x=1\), but since we have already canceled out the common factor of \((x-1)\), it does not create a vertical asymptote. Hence, the vertical asymptote of the function is x = 0.

Key concepts

Analytical Methods in Calculus

In the world of calculus, analytical methods are techniques we use to solve problems involving mathematical functions and their properties. This includes finding limits, derivatives, integrals, and asymptotes, which are all fundamental concepts in calculus.

When searching for vertical asymptotes analytically, we look for points where the function tends to infinity as it approaches a certain x-value. This usually happens when the denominator of a fraction approaches zero while the numerator does not. Identifying these values can help us understand the behavior of the function at its extremes. For example, in the exercise \(f(x)=\frac{x^{2}-3 x+2}{x^{10}-x^{9}}\), the analytical process involves factoring, simplifying, and investigating limits, which leads to the discovery of a vertical asymptote at \(x = 0\).

Factorization

Factorization is a powerful tool in algebra and calculus. It involves breaking down a complex expression into simpler components that can be easily analyzed or simplified. In calculus, factorization can significantly aid in finding limits, solving equations, or simplifying expressions.

For instance, factorizing the numerator \(x^2 - 3x + 2\) into \(x - 1)(x - 2)\) and the denominator \(x^{10} - x^{9}\) into \(x^9(x - 1)\) makes it clear that there's a common factor, which, when canceled out, simplifies the function. The canceled factors can sometimes indicate potential vertical asymptotes, but if they vanish from both the numerator and denominator, as seen with the \(x-1\) term in our exercise, they do not contribute to vertical asymptotes.

Limits and Continuity

Limits are fundamental to calculus and involve understanding the behavior of functions as they approach specific points. Continuity, on the other hand, ensures that the function's behavior has no interruptions or jumps.

In terms of vertical asymptotes, limits help us discern the points where the function grows without bounds. The presence of an asymptote implies the function is not continuous at that point, as it 'jumps' to infinity. In our exercise, by analyzing the limit of \(f(x)\) as \(x\) approaches zero, we confirm the presence of a vertical asymptote since the limit goes to infinity. This also teaches us that while a function might appear to have discontinuities at certain points due to common factors, thorough simplification and examination can reveal the true nature of its continuity.