Limits & Continuity Study Notes for GATE CSE

Limits

The limit of f(x) as x approaches the worth of a from the left is written.

and the limit of f(x) as x approaches the value of a from the right is written

Definition of a Limit at a Point

Therefore, if the left-hand limit does not equal the right-hand limit as x approaches a, then the limit as x approaches a does not exist.

Limits found Numerically and Algebraically

While most limits are often found graphically, as we've been discussing, it's not always practical or necessary if the function is defined algebraically.

For instance, say we are given that.  If we are looking for, we can find both the left- and right-hand limits by using tables.  By choosing x values that meet up with and closer to x = 3 from each side, we will analyze the behaviour of f(x)

Notice that when we chose values on either side of x = 3, they were values that were very on the brink of x = 3. It seems that as x approaches 3 from either side, the function values are approaching 6.  Therefore, it seems reasonable to conclude that.

Making tables can still be as time-consuming as graphing, so we'll use the subsequent rules to algebraically evaluate limits more efficiently. Most of those rules can intuitively be verified from watching the previously worked examples. 

Continuity

A function f(x) is continuous at x = a, if all of the following are true:

Continuity means that a function is continuous wherever the graph of the function has no holes, gaps, or jumps.  A function is claimed to be discontinuous at x = a, if a hole, gap, or break occurs within the graph at x = a, meaning the function violates one of the three items above.

Notice that the third item is within the definition of continuity and Rule.

Example.

Using the graph of f(x) below, find all values of x where f(x) is discontinuous and state why f(x) is discontinuous at these points, according to the definition of continuity.

Solution:

x = -3, f(-3) is undefined

x = -2, f(-2) is undefined

x = 1, while the function is defined by f(1)=5 and ,

these are unequal and thus the third item of the definition is violated

x = 4,  and does not exist.

An important formula of limit

Differentiability 

 Suppose y = f (x), then its derivative with respect to x, is commonly denoted by

 f ′(x) = y′ = D f (x) = D_x f (x)

The symbols and D are called differential operators.  They are used to explicitly denote the differentiation of the function that follows

Basic Differentiation Formulas:

Suppose f  and g are differentiable functions, c is any real number, then

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