Here is an example of a function:
It is an important example as students are likely to get confused as to whether
it is a function or not. As stated in the definition, since each value of x is associated
with exactly one value of y (though in this case that y is same for every x) hence
it is a function.
Example: Consider f to be defined by
This is a relation (not a function) since we can observe that 1 maps to 2 and 3,
Limit of a function:
Suppose f : R → R is defined on the real
line and p,L ∈ R. Then we say that the limit of the function
f is l if
For every real ε > 0, there exists a real δ > 0
such that for all real x,
0 < | x − p | < δ
implies | f(x) − L | < ε.
Mathematically it is represented as
Example: Find the limit of the function f(x) = (x2-6x
+ 8) / x-4, as x→5.
The limit is 3, because f (5) = 3 and this function is continuous at x
Such easy questions are not asked in the exam so it was just meant to clear the
concept of limit. We now move on to a bit difficult question:
Example: Find the limit of the function g(x) = √(x-4) -3 / (x-13)
as x approaches 13.
Solution: In such a question you first need to reduce the function
in simple form so that the computation of limit becomes simple.
First rationalize the numerator and denominator by multiplying by its conjugate
So, [√(x-4) -3] / (x-13) x [√(x-4) +3]/ [√(x-4) +3]
On multiplying and solving we get the result,
1/ √( x-4) +3
Now since the terms have been simplified the limit can now be calculated by substituting
the value of x as 13.
Hence putting x=13 in the last equation we get the limit as 1/6.
For more on limits, you may refer the video
Continuity of a function:
A function y = f(x) is continuous at point x=a
if the following three conditions are satisfied:
(1) f(a) is defined ,
(2) exists (i.e., is finite) ,
A function is continuous when its graph is a single unbroken curve. This definition
proves to be useful when it is possible to draw the graph of a function so that
just by the graph the continuity of the function can be judged.
Differentiation: Differentiation is an operation that allows us
to find a function that outputs the rate of change of one variable with respect
to another variable. The derivative of a function at a chosen input value describes
the rate of change of the function near that input value. The process of finding
a derivative is called differentiation. Geometrically, the derivative at a point
is the slope of the tangent line to the graph of the function at that point, provided
that the derivative exists and is defined at that point.
Since Differential Calculus is new to the students as they do not study it
in their 10th standard examination, so they are advised to master the topic by
practicing questions on Limits, Continuity and Differentiability. The
preparation of Differential Calculus also gives another opportunity to prepare
and revise the chapter on Functions, Sets and Relations. To get an idea about
the types of questions asked you may also consult the Previous Year Papers