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

Evaluate, `lim_(x to 2) (x^(3) - 8)/(x^(2) - 4)`

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

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  2. Evaluate lim(x to 4) (x^(2) - 7x + 12)/(x^(2) - 16)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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