How to solve limit continuity and discontinuity questions

I don’t know how to answer easiest ways but I’ll describe what are the general ways of solving limit problems.

Generally in the limit problem indeterminate forms are asked. If not you can directly plug in the value at given point where variable is tending in limit.

i. Standard Result 1

limx→axn−anx−a=nan−1 n is integer

Indeterminate form is 00at x=a.

There are question in which you can get these type of form or express them in this form.

Eg.

limx→ax23−a23x−a

ii. Limit at infinity:

Generally, the indeterminate forms are ∞−∞

General idea is to rationalize the numerator.

eg.

limx→∞x−−√−x−3−−−−√

iii. Standard Result 2:

limθ→0Sinθθ=1.

This helps to tackle with limit of trigonometric function.

thisfollows.limθ→0Tanθθ=1.

eg

limx→aSin(x−a)x2−a2

iv. Standard Results 3.

limx→0log(1+x)x=1

limx→0ex−1x=1

These results helps to take limit if it contains logarithmic and exponential functions.

eg

limx→2x−22log(x−1)

And at last but not the least, L’Hopital’s rule.

Let f and g be differentiable functions, with g'(x) not zero in an interval around a, except possibly at a itself. Also, one of the following must hold true:

both f(x) and g(x) have limit 0 as x approaches a.

both f(x) and g(x) have infinite limit (either positive or negative) as x approaches a.

Then, the limit of the ratio f(x)g(x) is equal to the limit of the ratio f′(a)g′(a) (where the prime indicates the appropriate derivative), as long as that limit exists, or is infinite