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### Calculus - Functions, Limits, Continuity - Problem Set I - domain, range, plots , graphs; Invertibility; L'Hospital's rule

Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level;
Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule.

## Outline of the Tutorial

### Some of the problems solved in this tutorial.

Finding the domains of the following functions.
(i) y = √(x 3 − x)
(ii) y = √(1 + 2sinx)
(iii) (x+3)/ √(x −5x+4)
(iv) y = sin−1 (log2 ( x2/2 ))

Drawing the graphs of the following functions.
(i) y = |x| + |x − 1|
(ii) y = [x], where [x] denotes the greatest integer not greater than x

Finding the domain and range of the function y = ln  (3x2 − 4x + 5).

Finding out if functions are invertible : Let f : R → R be deﬁned by f (x) = (ex −e -x)/2 , Is f (x) invertible? If so, ﬁnd its inverse.

Evaluating Limits:
lim (x2 − 16)/( x − 4)  as x->4
lim(x2 + x − 6) / (x2 - 4) as x->2

Evaluating the limits, as x-> 0, for : sin kx / x, tan x/ x ,  ( 1 - cos x)/ x , sin Ax / cos Bx and the limit for x ln x ( for x > 0)
Limits involving polynomials of 'x' in the numerator and denominator; limits involving series expansions, exponential and logarithmic functions.
A general familiarity with 0/0 and Infinity/Infinity kinds of limits.

... apart from this, we will also find the limits for a lot more interesting functions and we will also learn to apply L'Hospital's rule.

Complete Tutorial with examples, solutions, problems and plots of functions (Also try out the MCQ Quiz at the end):