Notice that 

        1 - 1/x           x - 1         x + sin x           x + 1            1 + 1/x      
                      =                <                    <                  =                    
(for large x)               
        1 + 1/x           x + 1        x + cos x          x - 1            1 - 1/x      

Now the left and right hand side both approach 1 as a approaches infinity.  The squeeze theorem implies that

       

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