Mathematics | Limits, Continuity and Differentiability

1. Limits –

For a function the limit of the function at a point is the value the function achieves at a point which is very close to .

Formally,
Let be a function defined over some interval containing , except that it
may not be defined at that point.
We say that, $L = \lim_{x\to a} f(x)$ if there is a number $\delta$ for every number $\epsilon$ such that
$|f(x)-L| < \epsilon$ whenever $0<|x-a|<\delta$

The concept of limit is explained graphically in the following image –

As is clear from the above figure, the limit can be approached from either sides of the number line i.e. the limit can be defined in terms of a number less that  or in terms of a number greater than . Using this criteria there are two types of limits –
Left Hand Limit – If the limit is defined in terms of a number which is less than  then the limit is said to be the left hand limit. It is denoted as $x\to a^-$ which is equivalent to  where .
Right Hand Limit – If the limit is defined in terms of a number which is greater than  then the limit is said to be the right hand limit. It is denoted as $x\to a^+$ which is equivalent to  where .

Existence of Limit – The limit of a function at exists only when its left hand limit and right hand limit exist and are equal i.e.
$\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x)$

Some Common Limits –

L’Hospital Rule –

Example 1 – Evaluate $\lim_{x\to 0} \frac{x\cos{(x)}-\sin{(x)}}{x^2\sin{(x)}}$

· Solution – The limit is of the form $\frac{0}{0}$ , Using L’Hospital Rule and differentiating numerator and denominator

=\:\lim_{x\to 0} \frac{\cos{(x)}-x\sin{(x)}-\cos{(x)}}{x^2\cos{(x)} + 2x\sin{(x)}}

· Example 2 – Evaluate $\lim_{x\to 0}\frac{a^{mx}-b^{nx}}{\sin {kx}}$

· Solution – On multiplying and dividing by and re-writing the limit we get –
$=\:\lim_{x\to 0}\frac{(a^{mx}-1) - (b^{nx}-1)}{kx}.\frac{kx}{\sin {kx}}$

2. Continuity –

· Example 1 – For what value of $\lambda$ is the function defined by

continuous at ?

· Solution – For the function to be continuous the left hand limit, right hand limit and the value of the function at that point must be equal.

2. Continuity –

3. Differentiability –