\(lim_{\alpha \rightarrow {\pi/4}}{sin \alpha -cos \alpha \over \alpha -{\pi\over 4}}\) is

(a)

\(\sqrt{2}\)

(b)

\(1\over \sqrt{2}\)

(c)

1

(d)

2

\(lim_{n \rightarrow \infty}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})\) is

(a)

\(1\over 2\)

(b)

0

(c)

1

(d)

\(\infty\)

\(lim_{x \rightarrow 0}{e^{sin \ x}-1\over x}=\)

(a)

1

(b)

e

(c)

\({1\over e}\)

(d)

0

\(lim_{x \rightarrow 0}{e^{tan \ x}-e^x\over tan x-x}=\)

(a)

1

(b)

e

(c)

\({1\over2}\)

(d)

0

The value of \(lim_{x\rightarrow k^-}x-\left\lfloor x \right\rfloor \) , where k is an integer is

(a)

-1

(b)

1

(c)

0

(d)

2

At x \(={3\over 2}\) the function \(f(x)={|2x-3|\over 2x-3}\) is

(a)

continuous

(b)

discontinuous

(c)

differentiable

(d)

non-zero

Let f :\(R \rightarrow R\) be defined by \(f(x)= \begin{cases}x & x \text { is irrational } \\ 1-x & x \text { is rational }\end{cases}\)  then f is

(a)

discontinuous at x = \({1\over 2}\)

(b)

continuous at  x = \({1\over 2}\)

(c)

continuous everywhere

(d)

discontinuous everywhere

The function \(f(x)= \begin{cases}\frac{x^{2}-1}{x^{3}+1} & x \neq-1 \\ P & x=-1\end{cases}\)is not defined for x = −1. The value of f(−1) so that the function extended by this value is continuous is

(a)

\({2\over3}\)

(b)

-\({2\over3}\)

(c)

1

(d)

0

Let f be a continuous function on [2, 5]. If f takes only rational values for all x and f(3) = 12, then f(4.5) is equal to

(a)

\({f(3)+f(4.5)\over 7.5}\)

(b)

12

(c)

17.5

(d)

\(f(4.5)-f(3)\over 1.5\)

Let a function f be defined by \(f(x)={x-|x|\over x}\) for x \(\neq\) 0 and f(0) = 2. Then f is

(a)

continuous nowhere

(b)

continuous everywhere

(c)

continuous for all x except x = 1

(d)

continuous for all x except x = 0