JEE Advanced Limits Continuity and Differentiability questions

8 Mathematics questions on Limits Continuity and Differentiability as asked in JEE Advanced, newest first.

Let f(x) be a continuously differentiable function on the interval (0,∞) such that f(1)=2 and displaystylelim_t→ xfract^10f(x)-x^10f(t)t^9-x^9=1 for each x>0. Then, for all x>0, f(x) is equal to
2024
Let k ∈ R. If displaystylelim_x → 0+ (sin(sin kx) + cos x + x )^2/x = e^6, then the value of k is
2024
Let f : R arrow R be a function defined by f(x) = begincases x^2 sin ((π)/(x^2) ), & if x ≠ 0, \ 0, & if x = 0. endcases Then which of the following statements is TRUE?
2024
Let S be the set of all (α, β) ∈ R × R such that lim_x → ∞ fracsin(x^2)(log_e x)^α sin (1/(x^2) )x^αβ(log_e(1+x))^β = 0. Then which of the following is (are) correct?
2024
Let the function f : R → R be defined by f(x) = dfracsin xe^π x dfrac (x^2023 + 2024x + 2025 ) (x^2 - x + 3 ) + dfrac2e^π x dfrac (x^2023 + 2024x + 2025 ) (x^2 - x + 3 ). Then the number of solutions of f(x) = 0 in R is…
2024
Let f : R → R and g : R → R be functions defined by f(x) = begincases x |x| sin (1/x ), & x ≠ 0, \ 0, & x = 0, endcases and g(x) = begincases 1-2x, & 0 ≤ x ≤ 1/2, \ 0, & otherwise. endcases Let a,b,c,d ∈ R. Define the…
2024
Let f:(0,1) → R be the function defined as f(x) = √(n) if x ∈ [1/(n+1), 1/n ) where n ∈ N. Let g:(0,1) → R be a function such that displaystyleint_x^2^x √((1-t)/t) dt < g(x) < 2√(x) for all x ∈ (0,1). Then…
2023
Let f:(0,1) → R be the function defined as f(x) = [4x] (x - 1/4 )^2 (x - 1/2 ), where [x] denotes the greatest integer less than or equal to x. Then which of the following statements is(are) true?
2023

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