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
- Evaluate: \[ \lim_{x \to 0} \frac{\sin 3x - 3 \sin x}{x^3} \]
- Given below are two statements: Statement I: $\lim_{x \to 0} \left( \frac{\tan^{-1} x + \log_e \sqrt{\frac{1+x}{1-x}} - 2x}{x^5} \right) = \frac{2}{5}$ Statement II: $\lim_{x \to 1} \left( \frac{2}{x^{1-x}} \right) = \frac{1}{e^2}$ In the light of the above statements, choose the correct answer from the options given below
- For $ t>-1 $, let $ \alpha_t $ and $ \beta_t $ be the roots of the equation $ \left( (t + 2)^{\frac{1}{7}} - 1 \right)x^2 + \left( (t + 2)^{\frac{1}{6}} - 1 \right)x + \left( (t + 2)^{\frac{1}{21}} - 1 \right) = 0. $ If $ \lim_{t \to 1^+} \alpha_t = a $ and $ \lim_{t \to 1^+} \beta_t = b $, then $ 72(a + b)^2 $ is equal to:
- If the function $ f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x} $ is continuous at $ x = 0 $, then $ f(0) $ is equal to:
- \(\frac{d}{dx} \left( e^{2x} + 2e^x \right)\)
Questions Asked in JEE Advanced exam
A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is:
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- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
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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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).