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The epsilon-delta (\( \varepsilon \)-\( \delta \)) definition is a cornerstone concept in calculus used to rigorously define what a limit is. It’s typically utilized to confirm a suspected limit by making precise what it means for a function to get ‘close’ to a particular value as the input approaches some value. In the case of a function approaching a limit as \( x \) goes to \( -\infty \), we're concerned with making the function values close to the limit by ensuring the input \( x \) is sufficiently far in the negative direction. The \( \varepsilon \) represents any small positive number indicating closeness.We must find an \( N \) such that when \( x < N \), the absolute difference between the function and the limit is less than the chosen \( \varepsilon \). Ensuring this helps prove that as the input moves sufficiently towards \( -\infty \), the function value increasingly approaches the limit. This exercise allowed us to dive into such reasoning by proving \( \lim_{x \rightarrow -\infty} \frac{x}{x+1} = 1 \).

"Limits at infinity" specifically refer to understanding what happens to a function's value as the input variable grows larger and larger in the positive or negative direction. These limits help describe the end behavior of functions. For \( \lim_{x \rightarrow -\infty} \frac{x}{x+1} = 1 \), the focus is on what the quotient \( \frac{x}{x+1} \) nears as x becomes negatively large. Essentially, as \( x \) decreases beyond all bounds, the expression \( x+1 \) in the denominator becomes much like \( x \) itself, simplifying the function to approximately \( \frac{x}{x} \) or 1. This section of the exercise serves as a showcase for looking at limits as x approaches infinitely large positive or negative values. It’s important to note the distinction between this and limits that approach specific finite values.

In calculus, a formal limit proof follows a structured approach using the \( \varepsilon \)-\( N \) definition to show definitively that a limit is reached under specific conditions. For \( \lim_{x \rightarrow -\infty} \frac{x}{x+1} = 1 \), constructing a formal proof requires:

• Simplifying the expression to find \( \frac{x}{x+1} - 1 \), which results in \( \frac{-1}{x+1} \) after simplification.
• Applying the condition \( \left| \frac{-1}{x+1} \right| < \varepsilon \) to notify closeness.
• Determining an appropriate \( N \) value where \( N = -1 - \frac{1}{\varepsilon} \), which ensures that for all \( x < N \), the inequality holds true.

This process confirms that no matter how small \( \varepsilon \) is, a corresponding threshold \( N \) can be found such that the limit holds firm as \( x \) continues its way to negative infinity. This methodology is not only vital for proving limits, but it also builds foundational understanding for more complex calculus concepts.