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Let alpha in R ,Prove that a function f ...

Let `alpha in R` ,Prove that a function `f : R rarr R` is differentiable at `alpha` , if and only if there is a function `g : R rarr R` which is continuous at `alpha` and satisfies `f(x) - f(alpha) - g(x)(x - alpha)` for all `x in R`

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PATHFINDER-LIMIT, CONTINUITY AND DIFFERENTIABILITY-QUESTION BANK
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  2. If x is a real number in [0, 1]. Then find the value of lim(m rarr inf...

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  3. Let alpha in R ,Prove that a function f : R rarr R is differentiable a...

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  4. Suppose p(x) = a0 + a1x + a2x^2 +…+ anx^n. If |p(x)| le |e^(x-1) - 1| ...

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  5. Let f(x)={((a(1-sinx)+b.cosx+5)/x^2,xlt0),(3, x=0),((1+((cx+dx^3)/x^...

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  6. about to only mathematics

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  7. Let f(x) be a continuous function in [-1, 1] and satisfies f(2x^2 - 1)...

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  8. Let g(x) =int0^x f(t) dt where f is such that 1//2lef(t)le1 for tin[0,...

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  9. Determine the values of x for which the following functions fails to b...

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  10. In a function f : [-2a, 2a] rarr R is an odd function such that f(x) =...

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  11. lim(xrarr0)sqrt(1-cos2x)/(sqrt2x) is equal to

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  12. lim(xrarr2)(-1)^x (where [x] is the greatest integer function ), is eq...

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  13. lim(xrarr1([x]+[x]) , (where [.] denotes the greatest integer function...

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  14. If zr=cos(ralpha/n^2)+i"sin"(ralpha/n^2), where r=1,2,3,…n, then lim(n...

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  15. lim((xrarrinfty)((x+1))^(x//3) is given by

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  16. If lim(xrarrinfty)(sqrt(x^2-x+1 )-ax-b) = 0 then the values of a and b...

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  17. lim(xrarr0)[(In cosx)/((1+x^2)^(1/4)-1))] is equal to

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  18. If lim(nrarrinfty)(an-frac(1+n^2)(1+n))=b a finite number then

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  19. The integer n for which lim(x rarr0)((cos x-1)(cosx-e^x))/x^n is a fin...

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  20. The value of lim(xrarrinfty)frac(x+cosx)(x+sinx is

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