Calculus Study Notes: Limits and Continuity

Lesson 2.3: Techniques for Computing Limits

Limits of Linear Functions

Linear functions have the form f(x) = mx + b. The limit of a linear function as x approaches a value a is simply the function value at a.

• Key Point: For linear functions, limx→a f(x) = f(a).

• Example: If f(x) = 2x + 7, then limx→3 f(x) = 2(3) + 7 = 13.

Theorem 2.1 (Limits of Linear Functions):

If f is a linear function, then for all a,

$\lim_{x \to a} f(x) = f(a)$

Limit Laws

Limit laws allow us to compute limits of more complex functions by breaking them into simpler parts.

• Sum Law: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

• Difference Law: $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$

• Constant Multiple Law: $\lim_{x \to a} [c f(x)] = c \lim_{x \to a} f(x)$

• Product Law: $\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• Quotient Law: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$

• Power Law: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$

• Root Law: $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$

Limits of Polynomial and Rational Functions

Limits of polynomials and rational functions can be found by direct substitution, provided the denominator is not zero.

• Polynomial Functions: $\lim_{x \to a} p(x) = p(a)$

• Rational Functions: $\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}$, provided $q(a) \neq 0$

• Example: $\lim_{x \to 2} (3x^2 + 4x + 2) = 3(2)^2 + 4(2) + 2 = 12 + 8 + 2 = 22$

Other Techniques for Evaluating Limits

When direct substitution leads to indeterminate forms (like 0/0), algebraic manipulation or equivalent expressions may be used to evaluate the limit.

• Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ can be simplified to $\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$

The Squeeze Theorem

The Squeeze Theorem is used when a function is bounded above and below by two functions that have the same limit at a point.

Theorem (Squeeze Theorem):

If $g(x) \leq f(x) \leq h(x)$ for all x near a (except possibly at a), and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

Trigonometric Limits

Two important trigonometric limits are:

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

Lesson 2.4: Infinite Limits

Definition of Infinite Limits

If f(x) grows arbitrarily large as x approaches a, we write $\lim_{x \to a} f(x) = \infty$. If f(x) decreases without bound, $\lim_{x \to a} f(x) = -\infty$.

• Example: $\lim_{x \to 0^+} \frac{1}{x} = \infty$

One-Sided Infinite Limits

One-sided infinite limits describe the behavior of a function as x approaches a from one side only (left or right).

• $\lim_{x \to a^+} f(x)$: x approaches a from the right

• $\lim_{x \to a^-} f(x)$: x approaches a from the left

Vertical Asymptotes

If $\lim_{x \to a} f(x) = \infty$ or $-\infty$, the line x = a is a vertical asymptote of f(x).

Lesson 2.5: Limits at Infinity

Limits at Infinity and Horizontal Asymptotes

Limits at infinity describe the behavior of a function as x becomes very large or very small. Horizontal asymptotes are lines y = L where $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$.

• Example: $\lim_{x \to \infty} \frac{1}{x} = 0$ (horizontal asymptote y = 0)

End Behavior of Polynomials and Rational Functions

The end behavior of a polynomial or rational function as x approaches infinity is determined by the highest degree term.

• Example: For $p(x) = 3x^4 - 6x^2 + x - 10$, as $x \to \infty$, $p(x) \sim 3x^4$

End Behavior for $e^x$, $e^{-x}$, and $\ln x$

• $\lim_{x \to \infty} e^x = \infty$

• $\lim_{x \to -\infty} e^x = 0$

• $\lim_{x \to \infty} \ln x = \infty$

• $\lim_{x \to 0^+} \ln x = -\infty$

Lesson 2.6: Continuity

Continuity at a Point

A function f is continuous at a point a if:

• f(a) is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Continuity on an Interval

f is continuous on an interval if it is continuous at every point in the interval. Right- and left-continuity are defined at endpoints.

Continuity of Composite, Polynomial, Rational, and Transcendental Functions

• Polynomials are continuous everywhere.

• Rational functions are continuous wherever the denominator is not zero.

• Composite functions are continuous if the inner and outer functions are continuous at the relevant points.

• Trigonometric, exponential, and logarithmic functions are continuous on their domains.

Intermediate Value Theorem (IVT)

If f is continuous on [a, b] and k is between f(a) and f(b), then there exists c in [a, b] such that f(c) = k.

Lesson 2.7: Precise Definitions of Limits

Epsilon-Delta Definition of a Limit

We say $\lim_{x \to a} f(x) = L$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that whenever $0 < |x - a| < \delta$, $|f(x) - L| < \varepsilon$.

• Key Point: This definition formalizes the concept of a limit using arbitrary closeness.

• Example: For $f(x) = 2x$, $\lim_{x \to 3} f(x) = 6$. For any $\varepsilon > 0$, choose $\delta = \varepsilon/2$.

Summary Table: Limit Laws

Law	Formula
Sum	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
Difference	$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
Constant Multiple	$\lim_{x \to a} [c f(x)] = c \lim_{x \to a} f(x)$
Product	$\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
Quotient	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ (if denominator ≠ 0)
Power	$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
Root	$\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$

Additional info: These notes include examples, theorems, and graphical illustrations to reinforce the concepts of limits and continuity, as well as their applications to polynomial, rational, trigonometric, exponential, and logarithmic functions. The Intermediate Value Theorem and the precise epsilon-delta definition of limits are also covered, providing a rigorous foundation for further study in calculus.