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P Pankaj Sanodiya
Given, Case 1: Limit at x = 0  The right-hand Limit or  Limit at  The left-hand limit or Limit at  Hence Limit will exist at x = 0 when m = n . Case 2: Limit at x = 1 The right-hand Limit or  Limit at  The left-hand limit or Limit at  Hence Limit at 1 exists at all integers.

P Pankaj Sanodiya
Given Now,

P Pankaj Sanodiya
Limit at x = a exists when the right-hand limit is equal to the left-hand limit. So, Case 1: when a = 0 The right-hand Limit or  Limit at  The left-hand limit or Limit at  Since Left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exists. Case 2: When a < 0  The right-hand Limit or  Limit at  The left-hand limit or Limit at  Since LHL = RHL,...

P Pankaj Sanodiya
Given, Now, Hence ,  Now, Hence  .

P Pankaj Sanodiya
Given, And   Since the limit exists, left-hand limit = Right-hand limit = f(1). Left-hand limit  = f(1) Right-hand limit From both equations, we get that,  and  Hence the possible value of a and b are 0 and 4 respectively.

P Pankaj Sanodiya
The right-hand Limit or  Limit at  The left-hand limit or Limit at  Since Left-hand limit and right-hand limit are equal, The limit of this function at x = 5 is 0.

P Pankaj Sanodiya
The right-hand Limit or  Limit at  The left-hand limit or Limit at  Since Left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exists.

P Pankaj Sanodiya
The right-hand Limit or  Limit at  The left-hand limit or Limit at  Since Left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exists.

P Pankaj Sanodiya
Limit at  Limit at  As we can see that Limit at  is not equal to Limit at ,The limit of this function at x = 1 does not exists.

P Pankaj Sanodiya
Given Function Now, Limit at x = 0  :   : Hence limit at x = 0 is 3. Limit at x = 1 Hence limit at x = 1 is 6.

P Pankaj Sanodiya
The function takes zero by zero form when the limit is put directly, so we simplify the function and then put the limits, So Let's put   Since we are changing the variable, limit will also change. as   So function in new variable becomes, As we know tha property       (Answer)

P Pankaj Sanodiya
On putting the limit directly the function takes infinity by infinity form, So we simplify the function and then put the limit    (Answer)

P Pankaj Sanodiya
The function takes the zero by zero form when we put the limit into the function directly, so we try to eliminate this case by simplifying the function. So   (Answer)

P Pankaj Sanodiya
As function doesn't create any abnormality on putting the limit directly,we can put limit directly. So, .  (Answer)

P Pankaj Sanodiya
The function takes the form zero by zero when we put the limit directly in the function,. since function consist of sin function and cos function, we try to make the function in the form of  as we know that it tends to 1 when x tends to 0. So,    (Answer)

P Pankaj Sanodiya
The limit:    The function takes the zero by zero form when the limit is put directly, so we simplify the function and then put the limit              (Answer)

P Pankaj Sanodiya
The limit the function behaves well on directly putting the limit,so we put the limit directly. So.    (Answer)

P Pankaj Sanodiya