The document considers whether the function f(x) = { is differentiable at x = 1. It uses the definition of the derivative as the limit of (f(x) - f(a)) / (x - a) as x approaches a from the left and right. It evaluates these one-sided limits and finds that the right-sided limit does not exist. Therefore, the function is not differentiable at x = 1.

1. Consider the alternate definition for the derivative of f(x) at x = a: f'(a) limx rightarrow a f(x) -
f(a)/x - a. Use limits from the left and the right to determine whether f(x) = { . is differentiable
at x = 1. Fully justify your answer.
Solution
limit f(x)-f(a) / ( x-a) = limit f(x)-f(a) / ( x-a) x->a+ x->a- for it to be
differentiable.at x=a f(1) = 1 as f(x) = x^2 for x<=1 let us find right limit limit f(x)-f(a) / ( x-a)
= limit f(1+h)-f(1) / ( (1+h)-1) x->1+ x->1+ f(1+h) = 2(1+h)-3 = (2h-1) as f(x) = 2x-3 for x>1
so this becomes limit (2h-1) - (1) / ( (1+h)-1) x->1+ =lt 2h-2/(h) which does not exist h-->0
right limit does not exist so this function is not differentiable at x=1