Calculus Study Notes: Limits and Graphical Analysis

Limits in Calculus

Definition and Properties of Limits

The concept of a limit is fundamental in calculus, describing the behavior of a function as its input approaches a particular value. Limits are used to define continuity, derivatives, and integrals.

• Limit Notation: $\lim_{x \to a} f(x)$ represents the value that f(x) approaches as x approaches a.

• Finite Limits: If f(x) approaches a real number as x approaches a, the limit is finite.

• Infinite Limits: If f(x) increases or decreases without bound as x approaches a, the limit is infinite ($+\infty$ or $-\infty$).

• Does Not Exist (DNE): If f(x) does not approach any particular value, the limit does not exist.

Evaluating Limits: Techniques and Examples

Several techniques are used to evaluate limits, including direct substitution, factoring, rationalization, and using special trigonometric limits.

• Direct Substitution: Substitute the value of x directly into f(x) if the function is continuous at that point.

• Factoring: Factor expressions to cancel terms and resolve indeterminate forms like $\frac{0}{0}$.

• Rationalization: Multiply by a conjugate to simplify expressions involving square roots.

• Special Trigonometric Limits:

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

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

Examples from Questions

• Example 1: $\lim_{x \to 2} \frac{3}{x^2}$

  • Direct substitution: $\frac{3}{2^2} = \frac{3}{4}$

• Example 2: $\lim_{x \to 0} \frac{\sin(x)}{\cos(x)}$

  • Direct substitution: $\frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0$

• Example 3: $\lim_{x \to 0} \frac{1}{x}$

  • As $x$ approaches 0, $\frac{1}{x}$ becomes unbounded. The sign depends on the direction of approach.

• Example 4: $\lim_{x \to 0^+} \frac{1}{x}$

  • Approaching 0 from the right, $\frac{1}{x} \to +\infty$

• Example 5: $\lim_{x \to \infty} \frac{1}{x^2}$

  • As $x$ increases, $\frac{1}{x^2} \to 0$

• Example 6: $\lim_{x \to 0} \frac{\sin(3x)}{4x}$

  • Use $\lim_{x \to 0} \frac{\sin(ax)}{bx} = \frac{a}{b}$: $\frac{3}{4}$

• Example 7: $\lim_{x \to 0} \frac{\cos(x)}{\sin(x)}$

  • As $x \to 0$, $\cos(x) \to 1$, $\sin(x) \to 0$. The limit does not exist (infinite).

• Example 8: $\lim_{x \to 1} 7x^2 + 2x + 1$

  • Direct substitution: $7(1)^2 + 2(1) + 1 = 7 + 2 + 1 = 10$

Special Limit Problem: Rationalization

Some limits require algebraic manipulation, such as rationalizing the numerator or denominator.

• Example: $\lim_{x \to 5} \frac{5 - \sqrt{x}}{x - 25}$

  • Direct substitution gives $\frac{5 - \sqrt{5}}{5 - 25} = \frac{5 - \sqrt{5}}{-20}$, but if $x = 25$, numerator and denominator are both zero, so rationalization is needed.

  • Multiply numerator and denominator by $5 + \sqrt{x}$ to simplify.

Graphical Analysis of Functions

Sketching Functions with Given Properties

Understanding how to sketch a function based on specific properties is essential in calculus. Properties may include function values, limits, derivatives, and points of discontinuity.

• Function Value: $f(-3) = 2$ means the graph passes through $(-3, 2)$.

• Infinite Limit: $\lim_{x \to -2} f(x) = \infty$ indicates a vertical asymptote at $x = -2$.

• Derivative Value: $f'(3) = -4$ means the slope of the tangent at $x = 3$ is $-4$.

• Function Value at Specific Points:

  • $f(1) = 3$

  • $f(4) = 5$

  • $f(0) = -4$

• Domain: $f$ is defined everywhere except maybe $x = -2$.

Steps to Sketch Such a Function

1. Plot the given points: $(-3, 2)$, $(1, 3)$, $(0, -4)$, $(4, 5)$.

2. Draw a vertical asymptote at $x = -2$ to represent the infinite limit.

3. Ensure the function is undefined at $x = -2$.

4. At $x = 3$, the tangent line should have a slope of $-4$.

Example Table: Properties of the Function

x	f(x)	Limit as x approaches	Derivative	Defined?
-3	2	-	-	Yes
-2	Undefined	$\infty$	-	No
0	-4	-	-	Yes
1	3	-	-	Yes
3	-	-	-4	Yes
4	5	-	-	Yes

Additional info: The above table summarizes the key properties required for sketching the function as described in the question.