91Ó°ÊÓ

The epsilon-delta definition of a limit is a formal way to describe the behavior of a function as it approaches a certain point. This definition is vital for understanding limits at infinity, like in the exercise.To say that a function \( f(x) \) approaches a limit \( L \) as \( x \) approaches infinity means:

• For every positive number \( \epsilon \) (however small), there exists a number \( N \) such that for all \( x < N \), the distance between \( f(x) \) and \( L \) is less than \( \epsilon \).

In essence, it guarantees that \( f(x) \) gets as close as we want to \( L \) if \( x \) becomes large enough.This definition helps us rigorously prove that \( lim_{x \to -\infty} \frac{x}{x+1} = 1 \). It introduces a systematic approach to managing how close function values get to their limits and involves precise handling of inequalities to determine the value \( N \) in the exercise.

Inequalities play a crucial role in limit problems, particularly in controlling the precision of our approximations.In the exercise, we had the inequality \( \left|f(x) - L\right| < \epsilon \).This inequality determines how close the function \( f(x) \) gets to the limit \( L \) of 1 for all values of \( x \) less than a certain point \( N \).By isolating the function inside the absolute value, we solidify the margin \( \epsilon \) by which \( f(x) \) approaches the limit.Thus, it becomes:

• \( \left| \frac{-1}{x+1} \right| < 0.001 \), simplifying to \( \frac{1}{|x+1|} < 0.001 \)

After manipulation, this converts to \( |x+1| > 1000 \), showing how inequalities guide us in evaluating how far \( x \) has to go towards \(-\infty\) for the limit condition to hold true.By managing these steps accurately, we define an appropriate value for \( N \).

The concept of the limit of a function is key to understanding how functions behave as their inputs grow very large or very small. In simple terms, a limit describes the value that \( f(x) \) "approaches" as \( x \) heads towards a particular point or infinity.For the limit \( \lim_{x \rightarrow -\infty} \frac{x}{x+1} = 1 \), it shows us that the output of the given function approaches 1 as \( x \) becomes increasingly negative.
To determine this, we assess the behavior of the ratio \( \frac{x}{x+1} \) at large negative \( x \) values, simplifying it into more comprehensible terms. The result indicates that the influence of the \(+1\) in the denominator becomes negligible as \( x \) decreases without bound. This gives insight into the limiting behavior of the function by illustrating how simpler components within \( f(x) \) shape the approach towards \( L \). Understanding limits allows one to predict function values and their behavior near infinity, a fundamental aspect of calculus.