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- Show by using an (ε − N ) argument that lim n->inf 3n^2 −2n+1/2n^2 − 4 =3/2
- Use an (ε − δ) argument to show that f : R → R be the function defined by
f(x)= {x^2−5x−5 if x≥−1 x^2+x+1 if x<−1}
is continuous at x = −1
f (x) be defined as follows
f(x) = {x^ 2+8x+15/x+3 ifx<−3
x^2 − 7 if x ≥ −3 }
Prove from first principles (i.e.an ε − δ proof) that f is continuous at the point x = −3.
Show that the function f :R -> R defined by f(x) = 2x+ 7 has an inverse by applying the inverse function theorem. Find its inverse also
Suppose that f :[0, 2] ->R is continuous on [0, 2] and differentiable on (0, 2), and that f (0) = 0, f (1) =1, f (2) =1. (i) Show that there exists c1 belongs to(0,1) such that f'(c1) =1 (ii) Show that there exists c2 belong to (1,2)such that f'(c2)=0(iii) Show that there exists c belongs to(0, 2) such that f'=1/3
An integrable function can have finitely many points of discontinuties. True or false with full explanation
Show that the function f defined by
F(x)=x^3+4x^2+x-6
has a real root in the interval [0,2]
Prove that the complement of every closed set is open.
1. a). Write down the definition (ε-δ language) of limx→x0 f(x) = L. b). Show that limx→+∞ cos x does not exist.
Let f be a differentiable function on [a,b ] and x belongs to[a,b]. Show that, if f'(x)=0 and f''(x)>0, then f must have a local maximum at x.
Let f :[0,1] tends to R be a function defined by f(x)=x^m (1-x)^n, where m,n belongs to N.Find the values of m and n such that the Rolle’s Theorem holds for the function f .
