Evaluate the following : Given that 7x ≤ f(x) ≤ 3x2 – 6 for all x. Determine the value of flimx→3f(x) - Mathematics and Statistics

Evaluate the following :

Given that 7x ≤ f(x) ≤ 3x² – 6 for all x. Determine the value of `lim_(x -> 3) "f"(x)`

बेरीज

It is given that

7x ≤ f(x) ≤ 3x² – 6 for all x

∴ `lim_(x -> 3) 7x ≤ lim_(x -> 3) "f"(x) ≤ lim_(x -> 3) (3x^2 - 6)`

∴ `7(3) ≤ lim_(x -> 3) "f"(x) ≤ 3(9) - 6`

∴ `21 ≤ lim_(x -> 3) "f"(x) ≤ 21`

By squeeze theorem, `lim_(x -> 3) "f"(x)` = 21.

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Concept of Limits

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

पाठ 7: Limits - Miscellaneous Exercise 7.2 [पृष्ठ १५९]

पाठ 7 Limits
Miscellaneous Exercise 7.2 | Q II. (6) | पृष्ठ १५९

Evaluate the following limit:

`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`

Evaluate the following limit :

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

Evaluate the following limit :

`lim_(y -> 1)[(2y - 2)/(root(3)(7 + y) - 2)]`

Evaluate the following limit :

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

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> 2) (x^2 - 1)` = 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 -> 1) f(x)` where `f(x) = {{:(x^2 + 2",", x ≠ 1),(1",", x = 1):}`

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 -> 0) sec x`

Sketch the graph of f, then identify the values of x₀ for which `lim_(x -> x_0)` f(x) exists.

f(x) = `{{:(sin x",", x < 0),(1 - cos x",", 0 ≤ x ≤ pi),(cos x",", x > pi):}`

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

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