 When x square is greater than equal to 1, continuous, differentiable at x equal to 1, differentiable but not continuous is equal to 1, this is absurd, differentiable but not continuous and that is not possible and continuous but not differentiable is equal to 1, it is continuous but not differentiable. What is the meaning of this? This is another way of one, this will be positive negative positive, you have to follow that in the same number, so basically from minus 1 to 1 I will stop like this, correct? This is 1, this is minus 1, I will not draw the x cube graph completely, I will only stop between minus 1 to 1. Now, when x is greater than 1, it is going to follow a straight line like this, right? When x is less than minus 1, it is again going to follow a straight line like this, clearly suggesting that there is a kink getting formed over here because of the abrupt change in the direction. But guys, never completely trust graphs, never completely trust graphs, I am saying that is because there are certain occasions where even though a curve has translated to a line, there is a smoothness over there, correct? If I talk of such a scenario where you had a problem, you did not deal with it, you know here it is differentiable actually, correct? That is what many people think from curve it became a line, so there has to be a kink getting formed hence not differentiable, no, you have to go out of micro level, exactly. If you see here the left hand derivative, this function is let us say x square function and this is let us say y equal to 0 function, this is y equal to x square function. The left hand derivative, what is the left hand derivative for this function at 0? Differentiate to it, of course continuity is there, so in this case it is differentiable. So my point here was, in this case it was very obvious but do not over rely on graph, do not over rely on the fact that there is a change on the curve from a curve to a line, so there has to be a kink there, correct? So in this case we need to always check, let us say I want to check mathematically whether there is a kink there or not, slightly left to 1, differentiate, so slightly left to 1 which function are you following? X cube, what is the derivative of X cube at 1? 3, what is the slope of this? 1. 1, hence there is a kink kink formula, is that right? It is same as 1 is 3, they are not equal hence not differentiate, is that right? Next question, option B is not good Let us do one thing, let us take 1 by X, 1 by X as negative y, now would you want minus y would tend to infinity, minus infinity or plus infinity? Minus infinity, plus infinity, because now all and 0 left is also negative, huge magnitude negative, the benefit of doing this is we can convert this to limit y tending to infinity, this will become negative 1 by y, e to the power 1 by x, limit x to the power n and tending to infinity, when mod x is less than 1 it will become 0, this one you are asking, limit x tending to x to the power n and tending to infinity, if mod of x is less than 1 then this will become 0, yes now tell me the answer for this, it will be 0, it will become 1 by 1 each, this will become, now this number So, your x is like 1 by e now, y does not become 0, 1 by e is approximately 0.3 something, if you raise it and when x minus infinity this will actually become 0, as x tends to 0, now when plus 0, again can I do one thing, let y be 1 by x, so I can write the same thing as, limit y tending to infinity, this will become 1 by y e to the power y minus e to the power minus y, y again e to the power y plus e to the power minus y, again if you take e to the power, what will happen, 1 by e to the power minus 2 y by 1 plus e to the power minus 2 y, that becomes again 1 by y because this term will go to 0, this term will go to 0 and this will since y is infinity, this will again give you 0 and does it match with the value of the function at 0, yes. So, those who said it is discontinuous at 0, the answer is no, it is continuous at x equal to 0, but is it differentiable at x equal to 0, let us check now, after that I would also like you to plot this on your desk boss and check, what it actually meant in terms of 0 and differentiable. Can we differentiate the point? Of course, you can left hand derivative, right hand derivative you can find out separately, but you know the best thing is use first principles here, you will be supplied, but that would be faster, should I show you the first principles, somebody please note this function now, because I am changing the page, should I complicate it, become very very small, yes. Now, what will be the basic definition minus h minus of f of x by h, so what I am doing is x with minus h, is it not, this is the definition h tending to 0 plus f of 0 minus h minus f of 0 by h. So, f of 0 minus h means what, f of minus h, so here I put x as minus h, so it will become minus h e to the power of minus 1 by h minus e to the power of plus 1 by h whole divided by minus 1 by h plus e to the power of 1 by h, correct, f of 0 is what, 0 and if I divided by h, in fact minus h, I will say, I will say, it will be this, yes sir, minus h minus h goes, now again the same story, I don't know h, h tending to 0 plus, what will happen to that, think carefully and then answer, e to the power of minus 1 by h as h tends to 0 will be 1, now 0, so if I take e to the power of 1 by h common out, from both the numerator and denominator, let me make the change here, this will become of 1, this will become minus 2, correct, so can I say this will be 0, this will be 0, this will get cancelled with each other, now in the same logic in your school exam and also mark it with a star, it's 1, 1, absolute, so limit h tending to 0 plus f of 0 plus h minus f of 0 by h, f of 0 plus h means replace x with h everywhere, is it fine, this is what you will, you must all be getting, yes, yes, now take e to the, if you do that you get h, 1 minus e to the power minus 2 by h, h, 1 plus e to the power minus 2 by h, what will happen to these terms, 0, 0, h, h gone, answer is 1, so it is not differentiable at 1, that means option number a is correct, it's continuous everywhere, but not differentiable at x equal to 0, somebody please put it on the graph and check, awesome, not differentiable at x, this is what she is getting, right, not differentiable at x, okay, are you now conscious, these things are something where you will be tested upon, so that property which I wrote there, limit x tending x to the power n, n tending to infinity and then what x is less than 1, that's helpful in many cases, let's take another one.