List of top Mathematics Questions on Continuity asked in JEE Main

The set of all values of aa for which lim⁡x→a([x−5]−[2x+2])=0\displaystyle\lim _{x \rightarrow a}([x-5]-[2 x+2])=0,where [∝][\propto] denotes the greatest integer less than or equal to ∝\propto is equal to

If the function f(x)={(1+∣cos⁡x∣)λ∣cos⁡x∣,0<x<π2μ,x=π2cot⁡6xecot⁡4xπ2<x<πf(x)=\begin{cases}(1+|\cos x|) \frac{\lambda}{|\cos x|} & , 0 < x < \frac{\pi}{2} \\\mu & , \quad x=\frac{\pi}{2} \\\frac{\cot 6 x}{e^{\cot 4 x}} & \frac{\pi}{2}< x< \pi\end{cases}is continuous at x=π2,then9λ+6log⁡eμ+μ6−e6λx=\frac{\pi}{2}, then 9 \lambda+6 \log _{ e } \mu+\mu^6- e ^{6 \lambda} is equal to

lim⁡n→∞12n(11−12n+11−22n+11−32n+…+11−2n−12n)\displaystyle\lim _{n \rightarrow \infty} \frac{1}{2^n}\left(\frac{1}{\sqrt{1-\frac{1}{2^n}}}+\frac{1}{\sqrt{1-\frac{2}{2^n}}}+\frac{1}{\sqrt{1-\frac{3}{2^n}}}+\ldots +\frac{1}{\sqrt{1-\frac{2^n-1}{2^n}}}\right) is equal to
Let f(x)={x2sin⁡(1x),x≠00,x=0f(x)=\begin{cases}x^2 \sin \left(\frac{1}{x}\right) & , x \neq 0 \\ 0 & , x=0\end{cases}Then at x=0x=0
If the function
f(x)= {  loge(1−x+x2)+loge(1+x+x2)sec⁡ x−cos⁡ x,   x∈(−π2,π2)−0  k, x=0 f(x) =   \begin{cases}    \frac {log_e(1−x+x^2)+log_e(1+x+x_2)}{sec⁡\ x−cos⁡\ x},      & \quad x∈(−\frac \pi2,\frac \pi2)−{0}\\     k,  & \quad x=0   \end{cases}
is continuous at x =0x = 0, then k is equal to

If for p ≠ q ≠ 0, the function
f(x)=p(729+x)−37729+qx3−9f(x) = \frac{{^{\sqrt[7]{p(729 + x)-3}}}}{{^{\sqrt[3]{729 + qx} - 9}}}
is continuous at x = 0, then