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Evalutate lim(x to 4)((x^(2) - x - 12)^(...

Evalutate `lim_(x to 4)((x^(2) - x - 12)^(18))/((x^(3) - 8x^(2) + 16x)^(9))`

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When x = 4, the expression `((x^(2) - x - 12)^(18))/((x^(3) - 8x^(2) + 16x)^(9))` takes the form `(0)/(0)`
`:. underset(x to 4)(lim) ((x^(3) - x - 12)^(18))/((x^(3) - 8x^(2) + 16x)^(9)) = underset(x to 4)(lim) ([(x - 4)(x + )]^(18))/([x(x^(2)-8x +16)]^(9)) = underset(x to 4)(lim) ([(x - 4)(x + 3)]^(18))([x(x-4)^(2)]^(9))`
`= underset(x to 4)(lim) ((x - 4)^(18) (x + 3)^(18))/(x^(9)(x - 4)^(18))`
`= underset(x to 4)(lim) ((x + 3)^(18))/(x^(9)) = ((4 + 3)^(8))/(4^(9)) = (7^(18))/(4^(9))`
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AAKASH INSTITUTE ENGLISH-LIMITS AND DERIVATIVES -Example
  1. Evaluate lim(x to 4) (x^(2) - 7x + 12)/(x^(2) - 16)

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  2. Evaluate, lim(x to 2) (x^(3) - 8)/(x^(2) - 4)

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  3. Evalutate lim(x to 4)((x^(2) - x - 12)^(18))/((x^(3) - 8x^(2) + 16x)^(...

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  4. Evaluate the following limits : Lim(xto1) (x-1)/(2x^(2)-7x+5)

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  5. Evaluate lim(x to 1) ((1)/(x^(2) + x - 2) - (x)/(x^(3) - 1))

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  6. Evaluate lim(x to sqrt(2)) (x^(9) - 3x^(8) + x^(6) - 9x^(4) - 4x^(2) ...

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  7. Evaluate : underset(X to 2) (x^(8) - 256)/(x - 2)

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  8. Evaluate lim(x to 3) (x^(7) - 2187)/(x^4 - 81)

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  9. Evaluate lim(x to a) ((x + 3)^(7//5) - (a + 3)^(7//5))/(x - a)

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  10. If lim(x to 3) (x^(n) - 3^(n))/(x - 3) = 1458 and n in N, find n.

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  11. If lim(x to a) (x^(7) + a^(7))/(x + a) = 7, find the value of a.

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  12. If lim(x to 1) (x^(3) - 1)/(x - 1) = lim(x to k) (x^(4) - k^(4))/(x^...

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  13. Evaluate lim(x to 0) (sqrt(x + 2) - sqrt(2))/(x)

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  14. Evaluate lim(x to 0) ("sin" 4x)/(6x)

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  15. Evaluate lim(x to 0) ("sin"^(2) (3x//2x))/(x)

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  16. Evaluate lim(x to0 ) ("sin" 3x)/("sin" 6x)

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  17. Evaluate the following limits : Lim(x to 0) (1- cos x)/(x^(2))

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  18. Evaluate lim(x to 0) (tan x + 4 tan 2x - 3tan 3x)/(x^(2) tan x)

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  19. Evaluate lim(x to (pi)/(4)) (1 - tanx)/(x - (pi)/(4))

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  20. Let f be a real-valued function defined by f(x) = 3x^(2) + 2x + 5 Find...

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