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Show that f(x)=cos x is continuous for a...

Show that f(x)=cos x is continuous for all values of x.

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To show that the function \( f(x) = \cos x \) is continuous for all values of \( x \), we need to demonstrate that for any point \( a \) in the real numbers, the following condition holds: \[ \lim_{x \to a} f(x) = f(a) \] ### Step-by-Step Solution: ...
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NAGEEN PRAKASHAN-Continuity and Differentiability-Miscellaneous Exercise
  1. Show that f(x)=cos x is continuous for all values of x.

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  2. Differentiate w.r.t. x the function. (3x^2-9x+5)^9

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  3. Differentiate w.r.t. x the functionsin^3x+cos^6x

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  4. Find dy/dx if y= (5x)^(3cos 2x)

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  5. sin^(-1)(xsqrt(x)),0 le x le 1

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  6. Differentiate w.r.t. x the function (cos^(-1)x/2)/(sqrt(2x+7)),-2<x<2

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  7. cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=(x)/...

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  8. (log x)^(log x),x gt1

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  9. Differentiate w.r.t. x the function cos" "(a" "cos" "x" "+" "b" "s in"...

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  10. (sinx-cosx)^((sinx-cosx)),(pi)/(4) ltx lt (3pi)/(4)

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  11. Differentiate w.r.t. x the function x^x+x^a+a^x+a^a ,for some fixed a...

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  12. Differentiate x^x^(2-3)+(x-3)^x^2 with respect to x :

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  13. Find (dy)/(dx),if y=12(1-cost), x=10(t-sint),-pi/2<t<pi/2

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  14. If y=sin^(- 1)x+sin^(- 1)sqrt(1-x^2) then find (dy)/(dx)

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  15. "If "xsqrt(1+y)+ysqrt(1+x)=0," prove that "(dy)/(dx)=-(1)/((x+1)^(2)).

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  16. If (x-a)^2+(y-b)^2=c^2, for some c > 0, prove that ([1+((dy)/(dx))^2]^...

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  17. If cosy=xcos(a+y), with cosa!=+-1, prove that (dy)/(dx)=(cos^2(a+y...

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  18. If x=a(cost+tsint) and y=a(sint-tcost), find (d^(2)y)/(dx^(2)).

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  19. If f(x)=|x|^3 , show that f"(x) exists for all real x and find it.

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  20. Using mathematical induction prove that d/(dx)(x^n)=n x^(n-1) for all...

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  21. Using the fact that s in (A + B) = s in A cos B + cos A s in Band the ...

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