91Ó°ÊÓ

The epsilon-delta definition is a fundamental concept in understanding the formal definition of a limit in calculus. It is a way of saying that you can make the values of a function very close to the limit by choosing values of the input (usually denoted as \(x\) or \(x < N\)) close enough to a specific point.
In the given exercise, we want to prove that the limit \( \lim _{x \rightarrow -\infty} \frac{4x-1}{2x+5} = 2 \) holds true using the epsilon-delta approach.
Here’s a breakdown:

• **Epsilon (\(\epsilon\)):** This is a small positive number that represents how close we want the function’s value \(|f(x) - L|\) to be to the limit \(L\) (which is 2 in this case).
• **Negative number (\(N\)):** This represents the input \(x\) from which the function values are guaranteed to stay within \(\epsilon\) of \(L\). In practical terms, for every \(\epsilon > 0\), there should exist an \(N\) such that for all \(x < N\), \(|f(x) - L| < \epsilon\).

For the problem at hand, we confirm this concept by deriving that \(N = -58\), meaning that as long as \(x\) is less than this value, the function \(\frac{4x-1}{2x+5}\) will be within 0.1 (\(\epsilon = 0.1\)) of 2 (\(L = 2\)).

Limits at infinity deal with the behavior of a function as the input \(x\) approaches positive or negative infinity. This is crucial when analyzing end-behavior of rational functions.
For the function \(\frac{4x-1}{2x+5}\), the limit at \(-\infty\) tells us that as \(x\) becomes increasingly negative, the function approaches a specific value, in this case, 2.
This happens because:

• When \(x\) is very large in magnitude (but negative), terms like \(-1\) and \(5\) become negligible compared to the terms involving \(x\), i.e., \(4x\) and \(2x\).
• The function simplifies to an expression dominated by these x-terms, specifically \(\frac{4x}{2x}\), which simplifies to \(2\).
• Thus, as \(x\) heads towards \(-\infty\), it reaches the horizontal asymptote of 2.

Understanding limits at infinity helps in sketching graphs of functions because it tells us about the horizontal asymptotes. This comprehension allows us to predict the function's behavior further along the x-axis.

Solving inequalities is a core mathematical skill used to determine the range of values satisfying a particular condition. In this exercise, we use inequalities to find values of \(x\) that keep the function \(\frac{4x-1}{2x+5}\) within an epsilon (\(\epsilon = 0.1\)) of the limit 2.
We express this with the inequality \(\left| \frac{-11}{2x+5} \right| < 0.1\), which simplifies as follows:

• Remove the absolute value, resulting in two inequalities: \(\frac{11}{2x+5} < 0.1\) and \(\frac{11}{2x+5} > -0.1\).
• These guide us to solve for \(x\) by manipulating each inequality:
• For \(\frac{11}{2x+5} < 0.1\), solve to find \(x > 52.5\).
• For \(\frac{11}{2x+5} > -0.1\), solve to find \(x < -57.5\).
• The critical solution for the condition \(x < N\) is from the inequality \(x < -57.5\), as we need a negative \(N\).

Thus, solving inequalities helps to find the bounds within which the original condition (i.e., function being close enough to the limit) is satisfied. In this case, choosing \(N = -58\) ensures all conditions meet.