LIMITS AND CONTINUITY

Limits

A limit describes the value that a function approaches as the input (or variable) approaches a particular point.

1. Notation:
   If f(x)  approaches a value Las x approaches c, we write:

           x tends c lim​f(x)=L

2. Key Ideas:

• Finite Limit: The function approaches a specific value L.

• Infinite Limit: The function grows without bound (lim ⁡x→c f(x)=∞).

• Limit from One Side:

  • Left-hand limit: lim ⁡x→c −f(x)

  • Right-hand limit: lim ⁡x→cf(x)

3. Existence of a Limit:
   A limit exists if and only if the left-hand limit equals the right-hand limit:
                                       lim⁡x→c−f(x)=lim⁡x→c+f(x)

  4.Special Cases:

• Indeterminate Forms: Situations like 0/0, ∞/∞, or 0.∞ require special techniques (e.g., L’Hôpital's Rule).

• Limits at Infinity: Analyzes the behavior of a function as x→∞or x→−∞

Continuity

A function f(x)is continuous at a point c if:

1. f(c) is defined.

2. lim ⁡x→c  f(x)  exists.

3. lim x→c ​f(x)=f(c).

If any of these conditions fail, the function is not continuous at ccc.

1. Types of Discontinuity:

   • Removable Discontinuity: A "hole" in the graph (e.g., f(c) is undefined but the limit exists).

   • Jump Discontinuity: The left-hand and right-hand limits exist but are not equal.

   • Infinite Discontinuity: The function approaches infinity at c.

2. Continuity on an Interval:

      A function is continuous on an interval if it is continuous at every point in that interval. 

LAB ASSESSMENT No:03