$\displaystyle \lim_{x \to 0}$ $\left(cos\,x+sin\,x\right)^{\frac{1}{x}}$ equals

$\displaystyle \lim_{x \to 0}$ $\left(cos\,x+sin\,x\right)^{\frac{1}{x}}\quad(1^{\infty}$ form $)$ $=e^{\displaystyle \lim_{x \to 0}\frac{\left(cos\,x+sin\,x-1\right)}{x}}$ $=e^{\displaystyle \lim_{x \to 0}\left[\frac{sin\,x}{x}-\left(\frac{1-cos\,x}{x}\right)\right]}$ $=e^{1-0}=e^{1}$

$\displaystyle \lim_{x \to 0}$ $\left(cos\,x+sin\,

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
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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 = $
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- limits and derivatives
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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:
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- limits and derivatives
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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 $.
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- Mathematics
- limits and derivatives
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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 $.
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- Mathematics
- limits and derivatives
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Notes on limits and derivatives

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:

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