Summary of "Calculus 1 Lecture 1.1: An Introduction to Limits"

Main Ideas and Concepts:

Introduction to Limits:
- Limits are foundational to Calculus and necessary for understanding how to calculate slopes of curves and areas under curves.
- The concept of Limits allows us to find the slope of a curve at a specific point and to determine the area under a curve.
Goals of Calculus:
- Goal 1: Find the slope of a curve at a point (Tangent Line).
- Goal 2: Calculate the area under a curve between two points.
The Tangent Problem:
- To find the slope of a curve at a point, we can approximate it using a Secant Line connecting two points on the curve.
- As one point (Q) approaches the other point (P), the Secant Line becomes a better approximation of the Tangent Line.
Definition of a Limit:
- A limit describes how close we can get to a point without actually reaching it, allowing us to understand the behavior of functions near that point.
- The limit is expressed as "Q approaches P," meaning Q gets infinitely close to P without being equal to it.
Finding Limits:
- The process of finding Limits often involves evaluating the function from both sides (left-hand limit and right-hand limit) and ensuring they converge to the same value for the limit to exist.
One-Sided Limits:
- The limit can be approached from the left or the right, and both must converge to the same value for the overall limit to exist.
- If they do not converge to the same value, the limit does not exist.
Special Cases:
- Limits can approach positive or negative infinity, indicating Vertical Asymptotes in the graph of the function.
Practical Application:
- The lecture includes examples of calculating Limits using tables and evaluating functions as they approach specific values.

Methodology/Instructions:

Finding the Slope of a Tangent Line:
1. Identify the point P on the curve.
2. Select a nearby point Q on the curve.
3. Calculate the slope of the Secant Line connecting P and Q.
4. Move Q closer to P and recalculate the slope.
5. The limit of the slope as Q approaches P gives the slope of the Tangent Line at point P.
Finding Limits:
1. Identify the value that the variable is approaching.
2. Create a table of values approaching that number from both sides.
3. Evaluate the function at these points to see what value it approaches.
4. Check if the left-hand limit and right-hand limit are equal.
5. If they are equal, that value is the limit; if not, the limit does not exist.