Question:
$\lim_{x \to \infty} \frac{3x^3 + 2x^2 - 7x + 9 }{4x^3 + 9x - 2 }$ is equal to
$\lim_{x \to \infty} \frac{3x^3 + 2x^2 - 7x + 9 }{4x^3 + 9x - 2 }$ is equal to
Updated On: Jul 28, 2022
- $\frac{2}{9}$
- $\frac{1}{2}$
- $\frac{-9}{2}$
- $\frac{3}{4}$
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The Correct Option is D
Solution and Explanation
$\displaystyle \lim _{x \rightarrow \infty} \frac{3 x^{3}+2 x^{2}-7 x+9}{4 x^{3}+9 x-2}$
$=\displaystyle \lim _{x \rightarrow \infty} \frac{x^{3}\left[3+\frac{2}{x}-\frac{7}{x^{2}}+\frac{9}{x^{3}}\right]}{x^{3}\left[4+\frac{9}{x^{2}}-\frac{2}{x^{3}}\right]}$
On putting $x \rightarrow \infty$, we get
$=\frac{[3 + 0 - 0 + 0]}{[4 + 0 -0]}=\frac{3}{4}$
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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).