PYQs Based Test - 03 : Limits, Continuity and Differentiability

A function f (x) is defined as f≤ft( x ) = ≤ft\{ {*20c e^x,&x < 1\\ x + ax^2+bx,&x ≥ 1 } ., where x ϵ R. Which one of the following statements is TRUE?

Consider the sequence xₙ= 0.5xₙ₋₁ + 1, n = 1, 2, ... ... with x₀ = 0. Then $\underset{n\to \infty }{\lim }{x}_n$

Let g≤ft( x ) = ≤ft\{ {*20c - x,&x ≤ 1\\ x + 1,&x ≥ 1 } . and f≤ft( x ) = ≤ft\{ {*20c 1 - x,&x ≤ 0\\ x^2,&x > 0 } ..

Consider the composition of f and g, i.e. (fog)(x) = f(g(x)). The number of discontinuities in (fog)(x) present in the interval (-∞, 0) is:

What is the value of $\underset{x\to 0y\to 0}{\lim }\frac{xy}{x^2+y^2}?$

The function y = |2 − 3x|