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This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}\underset{x\to 0^{-}}{lim}f\left(x\right)=\left|x\right|cos\frac{1}{x}\\ =\underset{h\to 0}{lim}\left|0-h\right|cos\frac{1}{\left(0-h\right)}=\underset{h\to 0}{lim}h.cos\frac{1}{h}=0\left[?cos\frac{1}{x}oscillatebetween-1and1\right]\\ \underset{x\to 0^{+}}{lim}f\left(x\right)=\left|x\right|cos\frac{1}{x}=\underset{h\to 0}{lim}\left|0+h\right|cos\frac{1}{\left(0+h\right)}=\underset{h\to 0}{lim}h.cos\frac{1}{h}=0\\ \underset{x\to 0}{lim}f\left(x\right)=0\\ \therefore \underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0}{lim}f\left(x\right)=0\\ Hence,f\left(x\right)iscontinuousatx=0.\end{array}$

Yes, the IELTS test fee is same for both Academic and General Training. Only the test questions are different, fee is not.

You can pay your IELTS fee online by credit card, debit card, net banking, or UPI. If you are booking offline, then you can pay by demand draft (DD) or cash at some IDP offices.

The Uttarakhand board 12th official mark sheet will be released by the UBSE. It will be given by the school to the students. The date and time of collecting the UK 12th mark sheet will be informed to the students. Students have to collect the 12th mark list from the schools. They should check the details on the UK 12th mark sheet. If any wrong info is found, they should inform school.

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}\underset{x\to 4^{-}}{lim}f\left(x\right)=\frac{\left|x-4\right|}{2\left(x-4\right)}\left[\begin{array}{l}forx<4,\left|x-4\right|=-\left(x-4\right)\\ forx>4,\left|x-4\right|=\left(x-4\right)\end{array}\right]\\ =\underset{h\to 0}{lim}\frac{-\left[4-h-4\right]}{2\left[4-h-4\right]}=\underset{h\to 0}{lim}\frac{h}{-2h}=-\frac{1}{2}\\ \underset{x\to 4^{+}}{lim}f\left(x\right)=\frac{\left|x-4\right|}{2\left(x-4\right)}=\underset{h\to 0}{lim}\frac{\left[4+h-4\right]}{2\left[4+h-4\right]}=\underset{h\to 0}{lim}\frac{h}{2h}=\frac{1}{2}\\ \underset{x\to 4}{lim}f\left(x\right)=0\\ \therefore \underset{x\to 4^{-}}{lim}f\left(x\right)\ne \underset{x\to 4^{+}}{lim}f\left(x\right)\ne \underset{x\to 4}{lim}f\left(x\right)\\ Hence,f\left(x\right)isdiscontinuousatx=4.\end{array}$

For the academic batch 2025, BTech aspirants can fill out the enquiry form at ANA Group of Institutions for the academic batch 2025. The college opens its admission portal for new applicants; candidates who meet the eligibility criteria according to the BTech course can apply for the course. 
Candidates are required to submit the details about results from Class 12 and the mentioned entrance exam in the application form. Moreover, candidates can get admission through management quota also. 

To calculate the marks in the NIPER JEE exam, applicants need to first identify the number of correct answers and incorrect answers. After that students need to multiply their correct number of answers with 0.5 marks and incorrect with 0.125 marks. After that students need to subtract the wrong answer marks from correct answers, and their final marks will be generated.

For IELTS courier booking, you need:

• Filled IELTS application form
• Clear passport copy (front and back page)
• A demand draft (DD) or other payment proof for exam fee

Put all papers safe and send to the IDP IELTS centre.

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}f\left(x\right)=\frac{2x^2-3x-2}{x-2}\\ =\frac{2x^2-4x+x-2}{x-2}=\frac{2x\left(x-2\right)+1\left(x-2\right)}{x-2}\\ =\frac{\left(2x+1\right)\left(x-2\right)}{x-2}=2x+1\\ \underset{x\to 2^{-}}{lim}f\left(x\right)=2x+1=\underset{h\to 0}{lim}2\left(2-h\right)+1=4+1=5\\ \underset{x\to 2^{+}}{lim}f\left(x\right)=2x+1=\underset{h\to 0}{lim}2\left(2+h\right)+1=4+1=5\\ \underset{x\to 2}{lim}f\left(x\right)=5\\ As\underset{x\to 2^{-}}{lim}f\left(x\right)=\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{x\to 2}{lim}f\left(x\right)=5\\ Hence,f\left(x\right)iscontinuousatx=2.\end{array}$

Yes, you can do IELTS slot booking by courier in India. You have to send your application form, passport copy, and payment details by post or courier to the IDP office.

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}\underset{x\to 0^{-}}{lim}f\left(x\right)=\frac{1-cos2x}{x^2}=\underset{h\to 0}{lim}\frac{1-cos2\left(0-h\right)}{{\left(0-h\right)}^2}\\ =\underset{h\to 0}{lim}\frac{1-cos\left(-2h\right)}{h^2}=\underset{h\to 0}{lim}\frac{1-cos\left(2h\right)}{h^2}\\ =\underset{h\to 0}{lim}\frac{2si{n}^{2}h}{h^2}\left[?1-cos\theta =2si{n}^2\frac{\theta }{2}\right]\\ =\underset{h\to 0}{lim}\frac{2sinh}{h}.\frac{sinh}{h}=2.1.1=2\left[\underset{x\to 0}{lim}\frac{sinx}{x}=1\right]\\ \underset{x\to 0^{+}}{lim}f\left(x\right)=\frac{1-cos2x}{x^2}\\ =\underset{h\to 0}{lim}\frac{1-cos2\left(0+h\right)}{{\left(0+h\right)}^2}=\underset{h\to 0}{lim}\frac{1-cos\left(2h\right)}{h^2}\\ =\underset{h\to 0}{lim}\frac{2si{n}^{2}h}{h^2}=\underset{h\to 0}{lim}\frac{2sinh}{h}.\frac{sinh}{h}=2.1.1=2\\ \underset{x\to 0}{lim}f\left(x\right)=5\\ As\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)\ne \underset{x\to 0}{lim}f\left(x\right)\\ Hence,f\left(x\right)isdiscontinuousatx=0.\end{array}$

Yes, look for books that say Digital SAT. The test adapts to you now. Harder questions come if you are smart. Kaplan explains this well. Old books dont know about modules. So avoid them.

The Faculty of Liberal Arts, MIT WPU secured the 96th rank in the Arts category of the Outlook-ICARE Rankings 2024, reflecting its strong academic foundation and presence among top liberal arts institutions in India.

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}\underset{x\to 2^{+}}{lim}f\left(x\right)=3x+5\\ =\underset{h\to 0}{lim}3\left(2+h\right)+5=11\\ \underset{x\to 2}{lim}f\left(x\right)=3x+5=3\left(2\right)+5=11\\ \underset{x\to 2^{-}}{lim}f\left(x\right)=x^2=\underset{h\to 0}{lim}{\left(2-h\right)}^2\\ =\underset{h\to 0}{lim}{\left(2\right)}^2+h^2-4h={\left(2\right)}^2=4\\ Since\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{x\to 2}{lim}f\left(x\right)\ne \underset{x\to 2^{-}}{lim}f\left(x\right)\\ Hence,f\left(x\right)isdiscontinuousatx=2.\end{array}$

Students need not worry if they fail in UK 12th exam 2026. They can appear for supplementary exam. The UK 12th Supplementary result 2026 will be out in October 2026 tentatively.

A total of 252 companies took part in the RK University placements 2025. Check out the table for reference:

Yes, you can go to IDP office or partner centre and do booking there. They will help you fill the form and pay the fee. But most students prefer online booking, it is faster.

Some the Top Recruiters for RK University are listed below: