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Let a(1),a(2),...,a(n) be fixed real nu...

Let `a_(1),a_(2),...,a_(n)` be fixed real numbers and let
`f(x)=(x-a_(1))(x-a_(2))(x-a_(3))...(x-a_(n)).`
Find `lim_(xtoa_(1))f (x).`
If `anea_(1),a_(2),..,a_(n),Compute lim_(xtoa) f(x).`

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AI Generated Solution

To solve the problem, we need to find the limits of the function \( f(x) = (x - a_1)(x - a_2)(x - a_3) \cdots (x - a_n) \) as \( x \) approaches \( a_1 \) and as \( x \) approaches another value \( a \) that is not equal to any of the \( a_i \). ### Step 1: Finding \( \lim_{x \to a_1} f(x) \) 1. **Substituting \( x = a_1 \)**: \[ f(a_1) = (a_1 - a_1)(a_1 - a_2)(a_1 - a_3) \cdots (a_1 - a_n) \] ...
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RS AGGARWAL-LIMIT-SOLVED EXAMPLES
  1. Evaluate lim(xrarr0)((x^(3)cotx)/(1-cosx)).

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  2. lim(x->0)(sin3x+7x)/(4x+sin2x)

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  3. Evaluate lim(x-> 0) (tan3x-2x)/(3x- sin^2 x)

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  4. Evaluate lim(xrarr0)(xtan4x)/(1-cos4x).

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  5. Evaluate lim(xrarr0)((1-cosxsqrt(cos2x)))/(x^(2)).

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  6. Evaluate lim(xrarr(pi)/(4))((sinx-cosx))/((x-(pi)/(4))).

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  7. Evaluate lim(xrarr(pi)/(2))(cosx)/(((pi)/(2)-x)).

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  8. Evaluate lim(xrarr(pi)/(6))((sqrt3sinx-cosx))/((x-(pi)/(6))).

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  9. Let f(x){:{(2x+3",",xle0),(3(x+1)",",xgt0.):} Find (i) lim(xrarr0)f(...

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  10. Let f(x)={:{(x^(2)-1",",xle1),(-x^(2)-1",",xgt1.):} "Find"lim(xrarr1)f...

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  11. Let f(x)={:{((|x|)/(x)",",xne0),(0",",x=0.):} Find lim(xrarr0)f(x).

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  12. Suppose f(x)={(a+bx, x<1), (4, x=1), (b-ax, x>1):} and if lim(xrarr1) ...

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  13. If f(x)={{:(|x|+1,xlt0),(0,x=0) ,(|x|-1,xgt0):} for what value (s) ...

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  14. Let f(x)={{:(mx^(2)+n",",xlt0),(nx+m",",0lexle1),(nx^(3)+m",",xgt1):} ...

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  15. Let a(1),a(2),...,a(n) be fixed real numbers and let f(x)=(x-a(1))(...

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  16. Let f(x)={x}= greater integer less than or eqal to x. For any integer ...

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  17. If is an odd function and underset(xto0)f(x) exists then prove tha...

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  18. If f is an even function, prove that lim(xto0^(-)) (x)=lim(xto0^(+)) ...

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  19. Show that lim(xto0^(-)) ((e^(1//x)-1)/(e^(1//x)+1)) does not exist.

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  20. Show that ("lim")(x->0)x/(|x|) does not exist.

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