If $ lim_{x \to 0 } \frac{ \{(a - n) nx - tan \, x \} sin \, nx }{x^2} = 0, $ where n is
non-zero real number, then a is equal to
- 0
- $\frac{n + 1}{n}$
- n
- $ n + \frac{ 1}{n}$
The Correct Option is D
Approach Solution - 1
$\Rightarrow lim_{ x \to 0 } \Bigg \{ (a - n) n - \frac{ tan \, x}{x} \Bigg \} \frac{ sin \, n \, x}{ n \, x} \times n = 0 $
$\Rightarrow $ $\hspace12mm$ { (a - n) n - 1 } n = 0
$\Rightarrow $ $\hspace12mm$ (a - n ) n = 1 $ \Rightarrow a = n + \frac{1}{n}$.
Approach Solution -2
Ans. The limit depicts how a function behaves at a certain point. The limit formula aids in the analysis of that function. The limit can therefore be described as:
"The behavior of some quantity dependent on an independent variable that tends or approaches a particular value"
- Limit in mathematics merely provides the output's high or close values.
- To calculate the derivatives, continuities, and integrals of functions, they are necessary.
- The equation is written as limx→a f(x) = b.
- According to the formulation above, the value of f(x) is equal to b if the limit approaches 'a.
Representations vary based on the type of limits. Here are some examples of it,
Right-hand side limits: It is represented as lim1 +f(x) = 1
Left-hand side limit: It is represented as lim1 -f(x) = 1
Infinite limits: In this f (x) value has no limit and it can extend to anywhere in the plane. It is represented as limx→∞ f(x) = 1
One-sided infinite limits: Here, one side of f(x) is represented as infinity. These are represented as, lim1 + f(x) = ∞ or lim1 – f(x) = ∞
Let y = f(x) be a function of x. If at some point x = a, f(x) takes an indeterminate form, we can take the values of the function that is close to a. If these values tend to a unique number like x tends to a, then the obtained unique number is known as the limit of f(x) at x = a.
Top Questions on Limits
- If \( f(x) = \begin{cases} \frac{(e^x - 1) \log(1 + x)}{x^2} & \text{if } x>0 \\ 1 & \text{if } x = 0 \\ \frac{\cos 4x - \cos bx}{\tan^2 x} & \text{if } x<0 \end{cases} \) is continuous at \( x = 0 \), then \(\sqrt{b^2 - a^2} =\)
- \(\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} =\)
- Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)
- Evaluate the limit: \[ \lim_{x \to 0} \frac{(\csc x - \cot x)(e^x - e^{-x})}{\sqrt{3} - \sqrt{2 + \cos x}} \]
- If a real valued function \( f(x) = \begin{cases} (1 + \sin x)^{\csc x} & , -\frac{\pi}{2} < x < 0 \\ a & , x = 0 \\ \frac{e^{2/x} + e^{3/x}}{ae^{2/x} + be^{3/x}} & , 0 < x < \frac{\pi}{2} \end{cases} \) is continuous at \(x = 0\), then \(ab = \)
Questions Asked in JEE Advanced exam
- Let $ \mathbb{R} $ denote the set of all real numbers. Let $ a_i, b_i \in \mathbb{R} $ for $ i \in \{1, 2, 3\} $.
Define the functions $ f: \mathbb{R} \to \mathbb{R},\ g: \mathbb{R} \to \mathbb{R},\ h: \mathbb{R} \to \mathbb{R} $ by:
$$ f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\quad g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4, $$ $$ h(x) = f(x+1) - g(x+2) $$ If $ f(x) \ne g(x) $ for every $ x \in \mathbb{R} $, then the coefficient of $ x^3 $ in $ h(x) $ is:- JEE Advanced - 2025
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Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
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Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
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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).