Evaluate the following limit : limx→1[x+x2+x3+.........+xn-nx-1]

Evaluate the following limit :

`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`

बेरीज

`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`

= `lim_(x -> 1)[(x + x^2 + x^3 + .... + x^"n" - (1 + 1 + 1 + ... "n times"))/(x - 1)]`

= `lim_(x -> 1) ((x - 1) + (x^2 - 1) + (x^3 - 1) + ... + (x^"n" - 1))/(x - 1)`

= `lim_(x -> 1)[(x^1 - 1^1)/(x - 1) + (x^2 - 1^2)/(x - 1) + (x^3 - 1^3)/(x - 1) + ... + (x^"n" - 1^"n")/(x - 1)]`

= 1 (1)⁰ + 2 (1)¹ + 3 (1)² + 4 (1)³ + ... + n (1)^n–1 ...`[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`

= 1 + 2 + 3 + 4 + ... + n

= `("n"("n" + 1))/2`

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पाठ 7: Limits - Exercise 7.1 [पृष्ठ १३९]

पाठ 7 Limits
Exercise 7.1 | Q III. (1) | पृष्ठ १३९

Evaluate the following limit:

`lim_(z -> -3) [sqrt("z" + 6)/"z"]`

Evaluate the following limit :

`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`

Evaluate the following limit :

`lim_(z -> "a")[((z + 2)^(3/2) - ("a" + 2)^(3/2))/(z - "a")]`

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 2) (x - 2)/(x^2 - 4)`

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	0.25641	0.25062	0.250062	0.24993	0.24937	0.24390

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 3) 1/(x - 3)`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> x/2) tan x`

Sketch the graph of a function f that satisfies the given value:

f(0) is undefined

`lim_(x -> 0) f(x)` = 4

f(2) = 6

`lim_(x -> 2) f(x)` = 3

Write a brief description of the meaning of the notation `lim_(x -> 8) f(x)` = 25

If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?