Let f, g and h be the real valued functions defined on ℝ as
f(x)={x/|x|,, x ≠ 0; 1,, x=0
g(x)={(sin (x+1))/(x+1),, x ≠-1; 1,, x=-1
and h(x)=2[x]-f(x), where [x] is the greatest integer ≤ x. Then the
value of lim(x → 1) g(h(x-1)) is :

Solution

Maths sir · 4 steps

Put t = x - 1 and check both sides of 0: h(t) sits at exactly -1, so you only need g(-1).

1. Shift the variable. Let t = x - 1. As x → 1, t → 0, so you need lim(t → 0) g(h(t)) with h(t) = 2[t] - f(t).

   Where t is	[t]	f(t)	h(t) = 2[t] - f(t)
   0 < t < 1	0	1	0 - 1 = -1
   -1 < t < 0	-1	-1	-2 + 1 = -1
   t = 0	0	1	0 - 1 = -1

2. h is constant. For every t in (-1, 1), h(t) = -1 exactly. It does not just approach -1. So near t = 0, g(h(t)) = g(-1).
3. Use the definition. At x = -1 the function g is given directly as g(-1) = 1. So g(h(t)) = 1 for all t near 0. (The limit of sin(x+1)/(x+1) as x → -1 is also 1, so there is no conflict.)
4. So. lim(x → 1) g(h(x-1)) = 1, which is option (a).

The idea to keep. In a limit of a composite function, first find what the inner function does near the point. If it equals a constant exactly, use the outer function's actual value there, not its limit.

Answer from the NTA final answer key. Solution by PrepWiser.