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Question

sin (T-x)15, lim

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Solution

Let the function be ,

f( x )= sin( πx ) π( πx )

We have to find the value of function at limit xπ .

So we need to check the function by substituting the value at particular point ( π ), so that it should not be of the form 0 0 .

If the condition is true, then we need to simplify the term to remove 0 0 form.

f( x )= sin( ππ ) π( ππ ) = 1 π sin0 0 = 1 π 0 0 = 0 0

Here, we see that the condition is not true and it is in 0 0 form.

The given expression f( x )= lim xπ sin( πx ) π( πx ) can be solved by assumption.

Let the value of ( πx ) be y . (1)

According the given limits of x ,

xπ , then y0

Also from equation 1, x=( πy )

On substituting the value of new limit in terms of y , the new expression is:

lim y0 siny ( πy ) (2)

According to the trigonometric theorem,

lim x0 sinx x =1 (3)

With the help of equations 2 and 3, we can calculate the value of limits;

lim y0 siny ( πy ) = lim y0 1 π siny ( y ) = 1 π lim y0 siny ( y )

On further simplification and using equation 3, we get

lim y0 siny ( πy ) = 1 π 1 = 1 π

Thus, the value of the given expression lim xπ sin( πx ) π( πx ) = 1 π


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