Introduction to Proofs Involving Limits

In calculus, limits are fundamental in understanding the behavior of functions as they approach specific points. Proofs involving limits help establish the validity of various mathematical statements and theorems. They are essential for confirming properties of functions, continuity, and differentiability. Here, we present three diverse and practical examples that illustrate different methods of proving limits.

Example 1: Proof of the Limit of a Polynomial Function

Context

Polynomial functions are continuous everywhere, but proving their limits can reinforce our understanding of how limits work in general.

When evaluating
[ \lim_{x \to 3} (2x^2 + 5x - 4) ]
we will demonstrate that we can find this limit directly by substitution.

To prove this, we substitute 3 directly into the polynomial:

[ \lim_{x \to 3} (2x^2 + 5x - 4) = 2(3^2) + 5(3) - 4 ]
[ = 2(9) + 15 - 4 ]
[ = 18 + 15 - 4 ]
[ = 29. ]

Thus, we conclude that
[ \lim_{x \to 3} (2x^2 + 5x - 4) = 29. ]

Notes

• This example illustrates that for polynomial functions, the limit can often be directly evaluated through substitution.
• Variations could include evaluating limits of other polynomials or exploring limits as x approaches infinity.

Example 2: Proof of the Limit of a Rational Function

Context

Rational functions can have limits that require simplification before evaluation. This example will show how to evaluate the limit of a rational function as it approaches a point where it may be indeterminate.

Consider the limit:
[ \lim_{x \to 1} \frac{x^2 - 1}{x - 1}. ]

First, we notice that substituting 1 directly gives us the indeterminate form ( \frac{0}{0} ). Thus, we will simplify the expression:

[ \frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1}. ]

For ( x \neq 1 ), we can cancel the ( (x - 1) ) terms:
[ = x + 1. ]

Now, we can evaluate the limit:
[ \lim_{x \to 1} (x + 1) = 1 + 1 = 2. ]

Thus,
[ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2. ]

Notes

• This example showcases the importance of simplifying rational functions before trying to evaluate limits.
• Variations could include exploring limits where the rational function has vertical asymptotes or evaluating limits at infinity.

Example 3: Proof of the Limit Using the epsilon-delta Definition

Context

The epsilon-delta definition of a limit is a rigorous way to prove limits. This method is particularly useful in formal proofs and higher mathematics.

Let’s prove that:
[ \lim_{x \to 2} (3x - 1) = 5. ]

We want to show that for every ( \epsilon > 0 ), there exists a ( \delta > 0 ) such that if ( 0 < |x - 2| < \delta ), then ( |(3x - 1) - 5| < \epsilon. )

Starting from the expression, we have:
[ |(3x - 1) - 5| = |3x - 6| = 3|x - 2|. ]

We need:
[ 3|x - 2| < \epsilon \Rightarrow |x - 2| < \frac{\epsilon}{3}. ]

Thus, we can choose ( \delta = \frac{\epsilon}{3} ). So, if ( 0 < |x - 2| < \delta ), we have:
[ |(3x - 1) - 5| < \epsilon. ]

This confirms that
[ \lim_{x \to 2} (3x - 1) = 5. ]

Notes

• This example emphasizes the formal approach to limits that underpins many calculus concepts and theorems.
• Variations could include proving limits for other linear functions or exploring nonlinear functions using the epsilon-delta definition.