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
For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \[ \lim_{x \to 0} \frac{x^2 \sin(\alpha x) + (\gamma - 1)e^{x^2}}{\sin(2x - \beta x)} = 3, \] then \( \beta + \gamma - \alpha \) is equal to:
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If $\lim_{x \to 1} \frac{(x-1)(6+\lambda \cos(x-1)) + \mu \sin(1-x)}{(x-1)^3} = -1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda + \mu$ is equal to
Questions Asked in JEE Advanced exam
- Consider a star of mass $ m_2 $ kg revolving in a circular orbit around another star of mass $ m_1 $ kg with $ m_1 \gg m_2 $. The heavier star slowly acquires mass from the lighter star at a constant rate of $ \gamma $ kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is $ r $, then its relative rate of change $ \frac{1}{r} \frac{dr}{dt} $ (in s$^{-1}$) is given by:
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The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced 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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As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ 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).