§ 3.1: Limits

Introduction to Limits

The concept of a limit is fundamental in calculus, serving as the basis for defining derivatives and integrals. Understanding limits allows us to analyze the behavior of functions as their inputs approach specific values, even if the function is not defined at those points.

• Limit of a Function: Let f be a function and a, L real numbers. We say that L is the limit of f(x) as x approaches a if, as x gets arbitrarily close to a (but not equal to a), f(x) gets arbitrarily close to L.

• Notation:

• If no such number L exists, the limit does not exist (abbreviated as d.n.e.).

One-Sided Limits

Sometimes, it is useful to consider the behavior of a function as x approaches a from only one side:

• Left-hand limit:

• Right-hand limit:

• Existence of Limit: For to exist, both one-sided limits must exist and be equal.

Evaluating Limits and Function Values

• The value of does not affect the existence or value of .

• Limits concern the behavior as x approaches a, not the value at a itself.

Example: Suppose a graph shows discontinuities or holes at certain points. You may be asked to find and , and , etc. The limit may exist even if the function is undefined at that point.

Formal (Epsilon-Delta) Definition of Limit

While not required for computation in this course, the formal definition provides mathematical rigor:

• We say if for every , there exists such that whenever , we have .

• Symbolically: such that

Additional info: This definition ensures that f(x) can be made as close as desired to L by choosing x sufficiently close to a.

Infinite Limits and Vertical Asymptotes

When f(x) increases or decreases without bound as x approaches a, we say the function has an infinite limit at a and a vertical asymptote at x = a.

• Notation: or

• These limits technically do not exist, but the notation describes the function's behavior.

• Formal definition: means for any , there exists such that if , then .

Limits at Infinity and Horizontal Asymptotes

As x becomes very large (positively or negatively), the function may approach a finite value L. In this case, the line y = L is a horizontal asymptote.

• Notation: or

• Formal definition: means for every , there exists such that if , then .

• Infinite limits at infinity: e.g., ,

Finding Limits at Infinity for Rational Functions

• If , then and (for defined values).

• For a rational function where and are polynomials:

  • If , then

  • If , then

  • If , then is the ratio of the leading coefficients of and

Example:

• (a)

• (b)

• (c)

Additional info: For (a), numerator degree > denominator, so limit is ; for (b), denominator degree > numerator, so limit is 0; for (c), degrees equal, so limit is ratio of leading coefficients: .

Properties of Limits

Suppose and exist. Then:

• Constant Rule:

• Constant Multiple Rule:

• Sum/Difference Rule:

• Product Rule:

• Quotient Rule: if

• Polynomial Continuity: If is a polynomial,

• Power Rule: For any , (if the limit exists)

• Equality Rule: If for all in some open interval containing , then

• Exponential and Logarithmic Functions: For any , ; for $b > 0$, ,

These properties are essential for computing limits efficiently.

Indeterminate Forms and Algebraic Manipulation

Sometimes, direct substitution in a limit yields an indeterminate form such as or . In these cases, algebraic manipulation is required to resolve the indeterminacy.

• Example:

• Example:

Additional info: For the first example, rationalizing the numerator or denominator can help; for the second, factoring out x inside the square root and simplifying is useful.