A particle starts at the origin and moves 1 unit horizontally to the right and reaches P₁, then it moves $${1 \over 2}$$ unit vertically up and reaches P₂, then it moves $${1 \over 4}$$ unit horizontally to right and reaches P₃, then it moves $${1 \over 8}$$ unit vertically down and reaches P₄, then it moves $${1 \over 16}$$ unit horizontally to right and reaches P₅ and so on. Let P_n = (x_n, y_n) and $$\mathop {\lim }\limits_{n \to \infty } {x_n} = \alpha $$ and $$\mathop {\lim }\limits_{n \to \infty } {y_n} = \beta $$. Then, ($$\alpha$$, $$\beta$$) is

The value of $$\mathop {\lim }\limits_{x \to {0^ + }} {x \over p}\left[ {{q \over x}} \right]$$ is

Let f : [a, b] $$ \to $$ R be differentiable on [a, b] and k $$ \in $$ R. Let f(a) = 0 = f(b). Also let J(x) = f'(x) + kf(x). Then

Let $$f(x) = 3{x^{10}} - 7{x^8} + 5{x^6} - 21{x^3} + 3{x^2} - 7$$.

Then $$\mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over {{h^3} + 3h}}$$