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Let us first draw the graph of y = x and y = tan x. <br> For `y = x, y' = 1`, i.e. the slope of the line is 1 for all real x. <br> For `y = tan x , y' = sec^(2)x` <br> Thus, both y = x and y = tan x have slope '1' at x = 0, hence the graphs touch each other. <br> Moving from '0' to `'pi//2'` the values of `sec^(2)x` increase from '1' to `oo`. <br> Thus, the graph pf y = tan x lies below the graph of y = x. <br> The graphs of y = x and y = tan x are plotted as follows. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/CEN_GRA_C03_E01_010_S01.png" width="80%"> <br> From the graph, when `x to 0^(+)`, the graph of y = tan x is above the graph of y = x <br> or `tan x gt x rArr (tan x )/(x) gt 1 rArr [underset(x to 0^(+))lim (tan x)/(x)] = 1` <br> When `x to 0^(-)`, the graph of y = x is above the graph of y = tan x <br> or `tan x lt x rArr (tan x )/(x) gt 1` (as x is negative) `rArr [underset(x to 0^(+))lim (tan x)/(x)] = 1` <br> Thus, `[underset(x to 0)lim (tan x)/(x)] = 1`Transcript

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00:00 - 00:59 | is evil way there is greatest integer limit extending 2010 x y x where this is represented by the greatest integer function right so basically this in this course you will find the value of this right and this thing can also quite as buying property of Limited can ride as limit X tends to zero and here are you are left with only 10 x 10 x y x Lite so we are going to use the result for solving this problem and cancer greatest integer problem which is can see that if limit X tends to zero sin x by X is less than one night is limit X tends to zero Sin inverse x y x it gives you greater than 1 write any two of us |

01:00 - 01:59 | and this is our second and third result is limit extending 2010 x divided by X is greater than 1 and forth result is limit extending 20 tan inverse x divided by X basically it is less than 1 write this 44 results with which is most of time are used in our question write where is given that in this course will find the value of greatest integer of limit X tends to zero and x divided by 3 which is basically tan limit X tends to zero tan x by X is greater than one night so we can say that this is value greater than 1 |

02:00 - 02:59 | greater than 18 gives you right to that's our final answer this question |