<p>The value of limx→0
−5 sin x
x
+
6 sin x
x
is (where [·] denotes G.I.F):</p>
Step-by-Step Solution
Key Concept: Recall that for x \to 0, sin x
x
approaches 1 from below (i.e., sin x
x
< 1).
<p><strong>1</strong>: Let sin x</p> x = 1 -ϵ.<p><strong>2</strong>: </p> -5 sin x x = [-5(1 -ϵ)] = [-5 + 5ϵ]. Since 5ϵ is slightly positive, the floor is -5.<p><strong>3</strong>: </p> 6 sin x x = [6(1 -ϵ)] = [6 -6ϵ]. Since 6ϵ is slightly positive, the floor is 5.<p><strong>4</strong>: Sum = -5 + 5 = 0.</p>
Correct Answer: 1