Differential Calculus - Limits and Continuity Three Marks Questions

11th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
    10 x 3 = 30
  1. Calculate \(\lim _{ x\rightarrow0}{|x| } \).

  2. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
    \(lim_{x\rightarrow{1}}sin \pi x\)

  3. The velocity in ft/sec of a falling object is modeled by \(r(t)=-\sqrt{32\over k}{1-e^{2t\sqrt{32k}}\over1+e^{-2r\sqrt{32k}}}\) ,where k is a constant that depends upon the size and shape of the object and the density of the air. Find the  limiting velocity of the object, that is, find \(lim_{t\rightarrow \infty}r(t).\)

  4. Find the left and right limits of \(f(x)={x^2-4\over (x^2+4x+4)(x+3)}at \ x=-2\) .

  5. Evaluate the following limits \(lim_{x\rightarrow\infty}{x^4-5x\over x^2-3x+1 }\)

  6. Evaluate \(\lim _{ x\rightarrow 2 }{ \frac { { x }^{ 3 }-8 }{ { x }^{ 2 }-4 } } \)

  7. Evaluate \(\lim _{ x\rightarrow 1 }{ \frac { \sqrt { { x }^{ 2 }-1 } +\sqrt { x-1 } }{ \sqrt { { x }^{ 2 }-1 } } } if\quad x>1\)

  8. Evaluate \(\lim _{ x\rightarrow 1 }{ \frac { (2x-3)\sqrt { x } -1 }{ { 2x }^{ 2 }+x-3 } } \)

  9. Evaluate \(\lim _{ x\rightarrow \pi }{ \frac { \sin { x } }{ x-\pi } } \)

  10. Examine the continuity of \(f\left( x \right) =\begin{cases} \frac { \sin { 2x } }{ \sin { 3x } } \quad if\quad x\neq 0 \\ 2\quad \quad \quad if\quad x=0 \end{cases}at\quad x=0\)

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