Question:
$\displaystyle\lim_{x \to\infty} \left(\frac{x^{2}}{3x-2}-\frac{x}{3}\right)=$
$\displaystyle\lim_{x \to\infty} \left(\frac{x^{2}}{3x-2}-\frac{x}{3}\right)=$
Updated On: Apr 19, 2024
- $\frac{1}{3}$
- $\frac{2}{3}$
- $\frac{-2}{3}$
- $\frac{2}{9}$
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The Correct Option is D
Solution and Explanation
Consider $\displaystyle \lim_{x \to\infty }\left[\frac{x^{2}}{3x-2}-\frac{x}{3}\right]$
$\displaystyle\ =lim_{x \to\infty } \left [\frac{3x^{2}-x\left(3x-2\right)}{3\left(3x-2\right)}\right]$
$=\displaystyle \lim _{x\to\infty} \frac{2x}{3\left(3x-2\right)} =\displaystyle\lim_{x\to\infty} \frac{2x}{3x\left[3-\frac{2}{x}\right]}$
$=\displaystyle \lim _{x\to\infty} \frac{2}{3} \frac{1}{3-\frac{2}{x}}=\frac{2}{3} \times\frac{1}{3-0}=\frac{2}{9}$
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).