 Hello and how are you all doing today? The question says evaluate the following limits. Now here the limit which is given to us is limit x approaches 0, sine a x plus b x upon a x plus sine b x, a b and a plus b are not equal to 0. Now before proceeding on we should be well versed that when limit x approaches 0 the value of limit sine x upon x is equal to 1, right? This is the key idea of this question. Proceeding on with the solution we are given in the question limit x approaches 0, sine a x plus b x upon a x plus sine b x. Now dividing the numerator and denominator by x limit x approaches 0, sine a x upon x plus b x upon x, the whole divided by a x upon x plus sine b x upon x, further on simplifying we have limit x approaches 0, sine a x upon x plus b the whole divided by a plus sine b x upon x. Now taking the limits to numerator and denominator separately we have limit x approaches to 0, sine a x upon x plus limit x approaches to 0, b divided by limit x approaches to 0, a plus limit x approaches to 0, sine b x upon x. Further now we know that x approaches to 0, a x also approach to 0. So we can write limit a x approaching 0, a sine a x upon a x we have multiplied the numerator and the denominator of sine a x upon x by a plus limit x approaches 0, b divided by limit x approaches 0, a plus limit b x will also approach to 0 as a x approach. So we have b sine b x upon b x. Now on using the limits we have the answer this limit as a into 1 plus plus b upon a plus b into this value will be 1 right. So we have a plus b upon a plus b which on simplification can be written as equal to 1. So 1 is the required answer to this question. Hope you understood the whole concept well. Have a very nice day ahead.