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
- Let \(f: R \to (0, \infty)\) be a twice differentiable function such that \(f(3) = 18\), \(f'(3)=0\) and \(f''(3) = 4\). Then \(\lim_{x \to 1} \log_e \left[ \frac{f(2+x)}{f(3)} \right]^{\frac{18}{(x-1)^2}}\) is equal to:
- Let \( f(x) = \int \frac{(2 - x^2) \cdot e^x{(\sqrt{1 + x})(1 - x)^{3/2}} dx \). If \( f(0) = 0 \), then \( f\left(\frac{1}{2}\right) \) is equal to :}
- The value of \[ \lim_{x\to 0}\frac{\log_e\!\big(\sec(ex)\cdot \sec(e^2x)\cdots \sec(e^{10}x)\big)} {e^2-e^{2\cos x}} \] is equal to:
- If the function \[ f(x)=\frac{e^x\left(e^{\tan x - x}-1\right)+\log_e(\sec x+\tan x)-x}{\tan x-x} \] is continuous at $x=0$, then the value of $f(0)$ is equal to
- If \[ \lim_{x\to 0} \frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}} {x\cos x-\log_e(1+x)} =2, \] then \(a^2+b^2+c^2\) is equal to
Questions Asked in JEE Advanced exam
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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).