Question

1. define infinity for someone who knows little about mathematics.
2. why do you think a removable discontinuity (hole) doesnt produce an asymptote on the graph of a polynomial function, even though it is excluded from the domain of the function?

Explanation:

Question 1

Infinity is a concept of something that has no end or limit. For example, if you start counting numbers: 1, 2, 3, 4... and you keep going, you'll never reach a last number. That idea of never being able to stop counting because there's always a next number is like infinity for counting numbers. Also, if you think about a line that goes on forever in both directions, it has infinite length—there's no point where it just stops, it keeps extending. In simple terms, infinity means something that is endless, without a boundary or a final quantity.

A removable discontinuity (hole) in a polynomial function occurs when there is a common factor in the numerator and denominator of a rational - like form of the polynomial (after factoring). For example, consider a function like \(f(x)=\frac{(x - 2)(x+3)}{(x - 2)}\) (where \(x
eq2\)). When we simplify this function (by canceling the common factor \((x - 2)\)), we get \(f(x)=x + 3\) (for \(x
eq2\)). An asymptote (like a vertical asymptote) occurs when the denominator approaches zero while the numerator does not approach zero (so the function values go to positive or negative infinity). In the case of a removable discontinuity, the factor that makes the denominator zero also makes the numerator zero (because of the common factor). So when we take the limit of the function as \(x\) approaches the value that is excluded from the domain (e.g., \(x = 2\) in our example), the limit exists (it's a finite number, like when \(x\) approaches 2 in \(f(x)=\frac{(x - 2)(x + 3)}{(x - 2)}\), the limit is \(2+3 = 5\)). Since the limit exists and is finite, the function values do not "blow up" to infinity (or negative infinity) near that point, so there is no asymptote. The hole is just a single point that is excluded from the domain, but the function behaves "nicely" (has a finite limit) around that point, unlike near a vertical asymptote where the function values become extremely large in magnitude.

Answer:

Infinity is a concept describing something that has no end, limit, or final quantity (e.g., the counting numbers 1, 2, 3… never end, or a line extending forever has infinite length).

Question 2