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If $ $ a $ $ is any integer, $ \mathop {\lim }\limits_{x \to \infty } \dfrac{{[ax + b]}}{x} $ is [where [.] denotes G.I.F]

Last updated date: 21st Jul 2024
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Hint: In this question we need to evaluate the value of the given function as $ x $ approaches infinity. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of $ x $ appearing in the denominator. Also we use the property of greatest integer function to evaluate this limit.

Complete step-by-step answer:
The greatest integer function is denoted by $ $ f\left( x \right){\text{ }} = {\text{ }}\left[ x \right] $ $ and is defined as the greatest integer less or equal to $ x $ . The greatest integer function of an integer is the integer itself, but for a non-integer value it will be an integer just before the given value. i.e.
 $ [5] = 5 $ , $ [2.8] = 2 $ .
Here they have given that is an integer, therefore $ [a] = a $
Consider, $ \mathop {\lim }\limits_{x \to \infty } \dfrac{{[ax + b]}}{x} $ first we will simplify this by taking common from the numerator and then dividing the common preset in both numerator and denominator,
 $ \mathop {\lim }\limits_{x \to \infty } \dfrac{{[ax + b]}}{x} = \mathop {\lim }\limits_{x \to \infty } \dfrac{{x\left[ {a + \dfrac{b}{x}} \right]}}{x} = \mathop {\lim }\limits_{x \to \infty } \left[ {a + \dfrac{b}{x}} \right] $
We know that in a fraction as the denominator becomes larger and larger the value of the fraction approaches zero, in the same way here also as $ x \to \infty $ the value $ \dfrac{b}{x} $ approaches zero i.e. $ \dfrac{b}{x} \to 0 $ $ x \to a $
Therefore $ \dfrac{{[ax + b]}}{x} \to a $ as $ x \to \infty $ .
Therefore the correct option is C.
So, the correct answer is “Option C”.

Note: Limits describe how a function behaves near a point, instead of at that particular point. The limit of a function $ f(x) $ at a point $ a $ in its domain is the value of the function as $ x $ approaches $ a $ to . The concept of a limit is the fundamental concept of calculus and analysis. It is used mainly to define continuity and differentiability, and it can also be used to analyze the local behavior of functions near points of interest.