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For the function f(x) = 4x. Find lim(x t...

For the function `f(x) = 4x`. Find `lim_(x to 2) f (x)`

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Let us compute the value of function for values of x very near to 2.
Firstly, we consider values of x approaching 2 from the left i.e., `x lt 2`
`{:(x,1.5,1.6,1.7,1.8,.1.9,1.99,1.999),(f(x)=4x,6,6.4,6.8,7.2,7.6,7.96,7.996):}`
Now, we consider values of x approaching 2 from the right i.e., `x gt 2`
`{:(x,2.3,2.2,2.1,2.05,2.03,2.01,2.001),(f(x)=4x,9.2,8.8,8.4,8.2,8.12,8.04.8.004):}`
Now from both the tables we observe that x approaches 2 either from left or right, the value of f(x) seems to approach 8.
So, `underset(x to 2)(lim) f(x) = underset(x to 2^(+))(lim) f(x) = underset(x to 2)(lim) f(x) = 8`
This can be shown by the graph also

Here, again the value of the function at x = 2 coincide with the limit at x = 2
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Knowledge Check

  • If the function f(x) satisfies lim_(x to 1) (f(x)-2)/(x^(2)-1)=pi , then lim_(x to 1)f(x) is equal to

    A
    1
    B
    2
    C
    0
    D
    3

  • If a differentiable function f(x) satisfies lim_(x to -1) (f(x) +1)/(x^(2)-1)=(3)/(2) Then what is lim_(x to -1) f(x) equal to ?

    A
    `-(3)/(2)`
    B
    `-1`
    C
    0
    D
    1

  • Let f :R to [0,oo) be such that lim_(xto3) f(x) exists and lim_(x to 3) ((f(x))^2-4)/sqrt(|x-3|)=0 . Then lim_(x to 3) f(x) equals

    A
    0
    B
    1
    C
    2
    D
    3
    • AAKASH INSTITUTE-LIMITS AND DERIVATIVES -Example
    1. For the function f(x) = 4x. Find lim(x to 2) f (x)

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    2. For the function f(x) = 2. Find lim(x to 1) f(x)

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    3. Find lim(x to 0) f(x), where f(x) = {{:(x -1,x lt 0),(0,x = 0),(x =1,...

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    4. Find (i) underset(x to 5)(lim 6) (ii) underset(x to 2)(lim x) (iii...

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    5. Evaluate lim(x to 2) [4x^(2) + 3x + 9]

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    6. Evaluate lim(x to 3)(x^(2) - 9)/(x + 2)

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    7. Evaluate lim(x to p) (sqrt(1 + x) + sqrt(1 - x))/(1 - x)

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

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

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

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    11. Evaluate underset(x to 1) (X - 1)/(2x^(2 - 7x + 5)

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

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

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

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

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

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

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

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

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

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