Concepts for Test (4.1-4.4)

Here's a breakdown of the concepts you need to know for the upcoming test covering sections 4.1 through 4.4:

Graphical Analysis
- Find a limit graphically: Given a graph of a function, determine the limit of the function as x approaches a specific value. This involves observing the behavior of the function as x gets closer and closer to the target value from both the left and the right.
- Identify point values graphically: Determine the value of the function at a specific point by reading the y-coordinate directly from the graph at the given x-coordinate.
- Determine limit existence: Decide whether a limit exists at a particular point. Remember, for a limit to exist, the function must approach the same y-value from both the left and the right sides of the point. If the left-hand limit and the right-hand limit are not equal, or if the function approaches infinity, the limit does not exist.
Continuity
- Determine whether a function is continuous: Check the three conditions for continuity at a point:
  1. The function must be defined at the point (i.e., f(a) exists).
  2. The limit of the function as x approaches the point must exist (i.e., \lim_{x \to a} f(x) exists).
  3. The limit must equal the function value at the point (i.e., \lim_{x \to a} f(x) = f(a)).
    If any of these conditions are not met, the function is discontinuous at that point.
Tangent Lines
- Find equations of tangent lines: Determine the equation of the line tangent to a curve at a given point. This involves finding the derivative of the function at that point to get the slope of the tangent line, and then using the point-slope form of a line to write the equation.
Analytical Limit Evaluation
- Find limits analytically: Evaluate limits using algebraic techniques.
  - Substitution: Try direct substitution first. If substituting the value directly into the function yields a determinate form, that is the limit.
  - Factoring: If direct substitution results in an indeterminate form (e.g., \frac{0}{0}), try factoring the numerator and/or denominator to simplify the expression and cancel out common factors.
  - Rationalization: If the function contains radicals, rationalize the numerator or denominator to eliminate the indeterminate form.
Limit Properties
- Apply properties of limits: Use the various properties of limits to simplify and evaluate limits. These properties include:
  - The limit of a sum is the sum of the limits.
  - The limit of a product is the product of the limits.
  - The limit of a quotient is the quotient of the limits (provided the limit of the denominator is not zero).
  - The limit of a constant times a function is the constant times the limit of the function.
Limits Involving Infinity
- Find limits at infinity and infinite limits:
  - Limits at infinity: Determine the behavior of a function as x approaches positive or negative infinity. This often involves dividing the numerator and denominator by the highest power of x and then evaluating the limit.
  - Infinite limits: Determine the behavior of a function as it approaches a specific value where the function increases or decreases without bound (approaches infinity). This often indicates the presence of a vertical asymptote.

If you have any questions as you study, please don't hesitate to ask!

Concepts for Test (4.1-4.4) Here's a breakdown of the concepts you need to know for the upcoming test covering sections 4.1 through 4.4: -

Graphical Analysis

Find a limit graphically: Given a graph of a function, determine the limit of the function as x approaches a specific value. This involves observing the behavior of the function as x gets closer and closer to the target value from both the left and the right. For example, if the function approaches a value L as x approaches a from both sides, then \lim_{x \to a} f(x) = L.
Identify point values graphically: Determine the value of the function at a specific point by reading the y-coordinate directly from the graph at the given x-coordinate. This is simply finding f(a) from the graph.
Determine limit existence: Decide whether a limit exists at a particular point. Remember, for a limit to exist, the function must approach the same y-value from both the left and the right sides of the point. If the left-hand limit and the right-hand limit are not equal, or if the function approaches infinity, the limit does not exist. Mathematically, \lim{x \to a^-} f(x) = \lim{x \to a^+} f(x)
Continuity
Determine whether a function is continuous: Check the three conditions for continuity at a point:

The function must be defined at the point (i.e., f(a) exists).
The limit of the function as x approaches the point must exist (i.e., \lim_{x \to a} f(x) exists).
The limit must equal the function value at the point (i.e., \lim_{x \to a} f(x) = f(a)).
If any of these conditions are not met, the function is discontinuous at that point.

Tangent Lines
Find equations of tangent lines: Determine the equation of the line tangent to a curve at a given point. This involves finding the derivative of the function at that point to get the slope of the tangent line, and then using the point-slope form of a line to write the equation. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is the point on the curve and m is the slope (derivative) at that point.
Analytical Limit Evaluation
Find limits analytically: Evaluate limits using algebraic techniques.
- Substitution: Try direct substitution first. If substituting the value directly into the function yields a determinate form, that is the limit. For example, \lim_{x \to 2} (x^2 + 1) = 2^2 + 1 = 5.
- Factoring: If direct substitution results in an indeterminate form (e.g., \frac{0}{0}), try factoring the numerator and/or denominator to simplify the expression and cancel out common factors. For example, \lim{x \to 1} \frac{x^2 - 1}{x - 1} = \lim{x \to 1} \frac{(x - 1)(x + 1)}{x - 1} = \lim_{x \to 1} (x + 1) = 2.
- Rationalization: If the function contains radicals, rationalize the numerator or denominator to eliminate the indeterminate form. For example, \lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x} can be solved by rationalizing the numerator.
Limit Properties
Apply properties of limits: Use the various properties of limits to simplify and evaluate limits. These properties include:
- The limit of a sum is the sum of the limits.
- The limit of a product is the product of the limits.
- The limit of a quotient is the quotient of the limits (provided the limit of the denominator is not zero).
- The limit of a constant times a function is the constant times the limit of the function.

For example:
\lim{x \to a} [f(x) + g(x)] = \lim{x \to a} f(x) + \lim{x \to a} g(x) \lim{x \to a} [f(x) \cdot g(x)] = \lim{x \to a} f(x) \cdot \lim{x \to a} g(x)
\lim{x \to a} \frac{f(x)}{g(x)} = \frac{\lim{x \to a} f(x)}{\lim{x \to a} g(x)}, if \lim{x \to a} g(x) \neq 0
\lim{x \to a} [c \cdot f(x)] = c \cdot \lim{x \to a} f(x)

Limits Involving Infinity

Okay, imagine you're playing with a toy car on a road, okay? That road is like a graph, and the toy car is like a function.

Graphical Analysis:

Finding a Limit: Imagine your car is going towards a certain point on the road. If, as your car gets super close to that point from the left and from the right, it seems to be heading towards the same spot, that's the limit! It's like where the car is "almost" going.
Point Values: If you stop the car at a certain spot on the road, the value at that point is just where the car is at that moment, easy peasy!
Limit Existence: Now, if the car is coming from the left side of the road and is heading to one spot, but if it comes from the right side, it's heading to a different spot, then there's NO limit! It's like the car can't decide where to go!

Continuity:

Imagine the road has no breaks, no jumps, and no holes. That means the road (or the function) is continuous! If there's a hole or a jump, it's not continuous.

Tangent Lines:

If you put a straight stick (tangent line) so it barely touches the road at one point, that stick shows you which way the car is pointing at that exact spot.

Analytical Limit Evaluation:

Substitution: Just plug in the number and see what you get!
Factoring: It's like simplifying a fraction. If you get 0/0, try to make the top and bottom simpler so you can cancel

Note