Find the limit of f(x) as x approaches infinity, and f(x) is in between two functions, and x is greater than 5.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) for x > 5 is defined such that `(4x - 1)/x < f(x) < (4x^2 + 3x)/x^2`

According to the Squeeze theorem for three functions g(x), f(x) and h(x) defined such that `g(x) <= f(x) <= h(x)` , if `lim_(x->a) g(x)` `= lim_(x->a) h(x) = L`   then `lim_(x->a) f(x) = L`

For `g(x) = (4x - 1)/x`

`lim_(x->oo) (4x - 1)/x = lim_(x->oo) 4 - 1/x = 4 - 0 = 4`

For `h(x) = (4x^2 + 3x)/x^2`

`lim_(x->oo) (4x^2 + 3x)/x^2 = lim_(x->oo) 4 - 3/x = 4 - 0 = 4`

Applying the squeeze theorem gives `lim_(x->oo) f(x) = 4`

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