Limits and Continuity Of Functions

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In limits and continuity, the concept of limits leads to define and describe continuity and derivative of the function. The continuity of a function has practical as well as theoretical importance. We plot graphs by taking the values generated in the laboratory or collected in the field. We connect the plotted points with a smooth and unbroken curve (continuous curve). This continuous curve helps as to estimate the values at the places where we haven't measured. It was developed by Isaac Newton and Leibnitz.

Here in this chapter, we will study some standard functions, their graphs, concept of limits and discuss about the continuity of the functions. Throughout this chapter, we denote R as the set of real numbers.

Types of Limits:

Left Hand Limit: Let f(x) tend to a limit l₁ as x tends to a through values less than 'a', then l₁ is called the left hand limit.

Right Hand Limit: Let f(x) tend to a limit l₂ as x tends to 'a' through values greater than 'a', then l₂ is called the right hand limit.

We say that limit of f(x) exists at x = a, if l₁ and l₂ are both finite and equal.

Limits of Trig Functions:

Limits of Trigonometric Functions and Sandwich Theorem:

for all x in some open interval containing c and suppose