Hello Friends now we see the problem number six which is also based on limits of exponential function and a logarithmic function of chapter limits the problem number six is a limit X tends to 0 3 raised X plus X minus 2 raised X plus 1 whole thing divided by sine X now let’s see the solution first of all we denote given problem as a L so it means L is equal to limit X tends to 0 3 raised X plus 5 raised X minus 2 raised X plus 1 whole thing divided by sin X so is equal to limit H tends to 0 now here 3 days X is a positive term so we have to write three days X minus 1 then again 5 a 6 is also positive term so we write plus 5 is X minus 1 and then the next term that to raise X plus 1 is negative now to balance is minus 1 minus 1 minus 2 so we have to add here plus 2 so this is require I just made in the numerator and whole thing divided by sine X now in the numerator we have to make a proper brackets that limit X tends to 0 we our trick is X minus 1 as a 1 bracket plus 5 is X minus 1 at the second bracket and then from the remaining last two terms we take minus 2 comment so in bracket we have two days X minus one and then totally divided by sine X now to get rule we have to divide numerator and denominator by X so the next step will be limit X tends to 0 3 days X minus 1 upon X as a 1 limit plus limit X tends to 0 5 is a X minus 1 upon X as a second limit and then minus 2 in bracket limit exchange to 0 to raise X minus 1 upon X at the thalamus and whole thing divided by limit exchange to 0 sine X upon X now limit exchange to 0 3 raise X minus 1 upon X that use answer log 3 plus limit extension 0 5 is X minus 1 upon X that use answer log minus 2 into limit extension leader to raise X minus 1 upon X that you’ll answer log 2 and whole thing divided by I’ll limit H tends to 0 sine X upon X that use 1 so is equal to log of 3 plus log of 5 minus now by using exponent law this I can write log to the whole square then that equal to log of 3 plus log of 5 minus log of 2 square 4 and that equal to log in bracket 3 into 5 upon soul and that equal to log of 15 upon soul so this is a required solution for the given problem thank you

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