## Limits, Continuity and Differentiability Questions and Answers Part-3

1. $\lim_{x \rightarrow 0}\frac{\sin\left(\pi\cos^{2}x\right)}{x^{2}}$     equals
a) $-\pi$
b) $\pi$
c) $\pi/2$
d) 1

Explanation:

2. For $x \epsilon R ,\lim_{x \rightarrow \infty}\left(\frac{x-3}{x+2}\right)^{x}=$
a) e
b) $e^{-1}$
c) $e^{-5}$
d) $e^{5}$

Explanation:

3. The integer n for which $\lim_{x \rightarrow0}\frac{(\cos x-1)\left(\cos x-e^{x}\right)}{x^{n}}$      is a finite nonzero number is
a) 1
b) 2
c) 3
d) 4

Explanation:

4. If f(x) = (x/2) – 1, then on the interval $\left[0,\pi\right]$
a) $\tan (f\left(x\right))$   and $\frac{1}{f\left(x\right)}$   are both continuous
b) $\tan (f\left(x\right))$   and $\frac{1}{f\left(x\right)}$   are both discontinuous
c) $\tan (f\left(x\right))$   and $f^{-1}\left(x\right)$   are both continuous
d) neither $\tan (f\left(x\right))$   nor $f^{-1}\left(x\right)$   is continuous

Explanation: f (x) = (x/2) - 1 is continuous on [0, $\pi$ ] and

5. The value of $\lim_{x \rightarrow \beta}\frac{1-\cos\left(ax^{2}+bx+c\right)}{\left(x-\beta\right)^{2}}$
where $\alpha,\beta$  are the distinct roots of $ax^{2}+bx+c=0$     is
a) $\left(\alpha-\beta\right)^{2}$
b) $\frac{\left(\alpha-\beta\right)^{2}}{2}$
c) $\left(\frac{a\left(\alpha-\beta\right)}{2}\right)^{2}$
d) none of these

Explanation:

6. $\lim_{x \rightarrow a}\left\{\left[\left(\frac{a^{1/2}+x^{1/2}}{a^{1/4}-x^{1/4}}\right)^{-1}-\frac{2\left(ax\right)^{1/4}}{x^{3/4}-a^{1/4}x^{1/2}+a^{1/2}x^{1/4}-a^{3/4}}\right]^{-1}-\sqrt{2}^{\log_{4}{a}}\right\}^{8}$
is
a) a
b) $a^{3/4}$
c) $a^{2}$
d) none of these

Explanation: Simplifying the expression in brackets by setting a1/4 = b and x1/4 = y, the function whose limit is required can be written as

7. The value of a for which $\frac{\sin 2x+a\sin x}{x^{3}}$    tends to a finite limits as $x\rightarrow 0$   is
a) 1
b) 2
c) -3
d) none of these

Explanation: Since numerator as well as denominator both

8. If g(x) is a polynomial satisfying g(x) g(y) = g(x) + g(y) + g(xy) -2 for all real x and y and g(2) =5 then $\lim_{x \rightarrow 3}$ g(x) is
a) 9
b) 25
c) 10
d) none of these

Explanation: Putting x = 2 and y = 1, we have
g(x) = xn+1 or g(x)

9. Let $f:R\rightarrow R$   be a function such that $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}$     for all x and y , and f(0)= 3 and f'(0) = 3 then
a) $f\left(x\right)$ /x is continuous on R
b) $f\left(x\right)$  is continuous on R
c) $f\left(x\right)$  is bounded on R
d) none of these

10. Given the function f(x) = 1/(1 – x), the number points of discontinuity of the composite function $y=f^{3n}\left(x\right),$   where $f^{n}\left(x\right)=fof$    .... of (n times ) are $\left(n\epsilon N\right)$