Question:
If $G(x) = - \sqrt{25 - x^2}$ then $\displaystyle \lim_{x \to 1} \frac{G(x) - G(1)}{x - 1}$ has the value
If $G(x) = - \sqrt{25 - x^2}$ then $\displaystyle \lim_{x \to 1} \frac{G(x) - G(1)}{x - 1}$ has the value
Updated On: Sep 19, 2024
- $\frac{1}{24}$
- $\frac{1}{5}$
- $ - \sqrt{24}$
- none of these
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
$\displaystyle\lim_{x \to1} \frac{- \sqrt{25-x^{2}} - \left(-\sqrt{24}\right)}{x-1}$
$ = \displaystyle\lim_{x \to1} \frac{\sqrt{24} - \sqrt{25-x^{2}}}{x-1} \times\frac{\sqrt{24} + \sqrt{25-x^{2}}}{\sqrt{24} + \sqrt{25-x^{2}}} $
$= \displaystyle\lim_{x \to1} \frac{x^{2} - 1}{\left(x-1\right) \left[\sqrt{24} + \sqrt{25 -x^{2}}\right]}$
$ = \displaystyle\lim_{x \to1} \frac{x+1}{ \left[\sqrt{24} + \sqrt{25-x^{2}}\right]} $
$= \frac{2}{2\sqrt{24}} = \frac{1}{2\sqrt{6}} $
$ = \displaystyle\lim_{x \to1} \frac{\sqrt{24} - \sqrt{25-x^{2}}}{x-1} \times\frac{\sqrt{24} + \sqrt{25-x^{2}}}{\sqrt{24} + \sqrt{25-x^{2}}} $
$= \displaystyle\lim_{x \to1} \frac{x^{2} - 1}{\left(x-1\right) \left[\sqrt{24} + \sqrt{25 -x^{2}}\right]}$
$ = \displaystyle\lim_{x \to1} \frac{x+1}{ \left[\sqrt{24} + \sqrt{25-x^{2}}\right]} $
$= \frac{2}{2\sqrt{24}} = \frac{1}{2\sqrt{6}} $
Was this answer helpful?
2
1
Top Questions on limits and derivatives
Let $\left\lfloor t \right\rfloor$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which
\[ \lim_{x \to 0^+} \left( x \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \dots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \dots + \left\lfloor \frac{9^2}{x^2} \right\rfloor \right) \geq 1 \]
is equal to __________.
- JEE Main - 2025
- Mathematics
- limits and derivatives
- Let the function \( g: (-\infty, 0) \rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) be given by \( g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2} \). Determine the properties of \( g \).
- BITSAT - 2024
- Mathematics
- limits and derivatives
- Let \( f \) be the function defined by:
\[ f(x) = \begin{cases} \frac{x^2 - 1}{x^2 - 2|x-1| - 1}, & \text{if } x \neq 1, \\ \frac{1}{2}, & \text{if } x = 1. \end{cases} \] The function is continuous at:- BITSAT - 2024
- Mathematics
- limits and derivatives
- Let \( f(x) = x^2 \log(\cos x) \log(1 + x) \) for \( x \neq 0 \), and \( f(0) = 0 \). Determine the behavior of \( f(x) \) at \( x = 0 \).
- BITSAT - 2024
- Mathematics
- limits and derivatives
- If \( f(x) \) is defined as follows:
\[ f(x) = \begin{cases} 4, & {if } -\infty < x < -\sqrt{5}, \\ x^2 - 1, & {if } -\sqrt{5} \leq x \leq \sqrt{5}, \\ 4, & {if } \sqrt{5} \leq x < \infty. \end{cases} \] If \( k \) is the number of points where \( f(x) \) is not differentiable, then \( k - 2 = \)- BITSAT - 2024
- Mathematics
- limits and derivatives
View More Questions
Questions Asked in JEE Advanced exam
- Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
- JEE Advanced - 2024
- Matrices
- A positive, singly ionized atom of mass number \( A_M \) is accelerated from rest by the voltage \( 192 \, \text{V} \). Thereafter, it enters a rectangular region of width \( w \) with magnetic field \( \vec{B}_0 = 0.1\hat{k} \, \text{T} \). The ion finally hits a detector at the distance \( x \) below its starting trajectory. Which of the following option(s) is(are) correct? \[ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} \]
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) pointing in the \( +z \)-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height \( L \), is placed in the x-y plane with its vertex \( P \) at \( x = 0 \) in the orientation shown in the figure. At \( t = 0 \), the loop starts entering the region of the magnetic field with a uniform velocity \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf (\( \mathcal{E} \)) in the loop as a function of the distance (\( x \)) starting from \( x = 0 \)?- JEE Advanced - 2024
- Radioactivity
- A table tennis ball has radius \( \frac{3}{2} \times 10^{-2} \, \text{m} \) and mass \( \frac{22}{7} \times 10^{-3} \, \text{kg} \). It is slowly pushed down into a swimming pool to a depth \( d = 0.7 \, \text{m} \) below the water surface and then released from rest. It emerges from the water surface at speed \( v \), without getting wet, and rises to a height \( H \). Which of the following option(s) is(are) correct?\(\text{(Given: } \pi = \frac{22}{7}, g = 10 \, \text{m/s}^2, \text{density of water} = 1 \times 10^3 \, \text{kg/m}^3, \text{viscosity of water} = 1 \times 10^{-3} \, \text{Pa.s).}\)
- JEE Advanced - 2024
- Radioactivity
View More Questions
Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
Properties of Derivatives:
Read More: Limits and Derivatives