Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Discuss the continuity of the following functions:
  1. $\text{f(x)}=\sin\text{x}+\cos\text{x}$
  2. $\text{f(x)}=\sin\text{x}-\cos\text{x}$
  3. $\text{f(x)}=\sin\text{x}\cos\text{x}$
✓

Answer

It is know that if g and h are two continuous functions, then
g + h, g - h, and g, h are also continuous.
It has to proved first that $\text{g(x)}=\sin\text{x}$ and $\text{h(x)}=\cos\text{x}$ are continuous functions.
Let $\text{g(x)}=\sin\text{x}$
It is evident that $\text{g(x)}=\sin\text{x}$ is defined for every real number.
Let c be a real number. Put x = c + h
If x → c, then h → 0
$\text{g(c)}=\sin\text{c}$
$\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}\sin\text{x}$
$=\lim\limits_{\text{h}\rightarrow0}\sin(\text{c}+\text{h})$
$=\lim\limits_{\text{h}\rightarrow0}\big[\sin\text{c}\cos\text{h}+\cos\text{c}\sin\text{h}\big]$
$=\lim\limits_{\text{h}\rightarrow0}(\sin\text{c}\cos\text{h})+\lim\limits_{\text{h}\rightarrow0}(\cos\text{c}\sin\text{h})$
$=\sin\text{c}\cos0+\cos\text{c}\sin0$
$=\sin\text{c}+0$
$=\sin\text{c}$
$\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{g(x)}=\text{g(c)}$
Therefore, g is a continuous function.
Let $\text{h(x)}=\cos\text{x}$
It is evident that $\text{h(x)}=\cos\text{x}$ is defined for every real number.
Let c be a real number. Put x = c + h
If x → c, then h → 0
$\text{h(c)}=\cos\text{c}$
$\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\lim\limits_{{\text{x}}\rightarrow\text{c}}\cos\text{x}$
$=\lim\limits_{{\text{h}}\rightarrow0}\cos(\text{c}+\text{h})$
$=\lim\limits_{{\text{h}}\rightarrow0}\big[\cos\text{c}\cos\text{h}-\sin\text{c}\sin\text{h}\big]$
$=\lim\limits_{{\text{h}}\rightarrow0}\cos\text{c}\cos\text{h}-\lim\limits_{{\text{h}}\rightarrow0}\sin\text{c}\sin\text{h}$
$=\cos\text{c}\cos0-\sin\text{c}\sin0$
$=\cos\text{c}\times1-\sin\text{c}\times0$
$=\cos\text{c}$
$\therefore\ \lim\limits_{{\text{x}}\rightarrow\text{c}}\text{h(x)}=\text{h(c)}$
Therefore, h is a continuous function.
Therefore, it can be concluded that
  1. $\text{f(x)}=\text{g(x)}+\text{h(x)}=\sin\text{x}+\cos\text{x}$ is a continuous function.
  2. $\text{f(x)}=\text{g(x)}-\text{h(x)}=\sin\text{x}-\cos\text{x}$ is a continuous function.
  3. $\text{f(x)}=\text{g(x)}\times\text{h(x)}=\sin\text{x}\times\cos\text{x}$ is a continuous function.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the particular solution of the differential equation
$(1 -\text{y}^{2})(1 + \log x) \text{dx + 2xy dy} = \text{0, given that y = 0 when x = 1.} $
→
Evaluate the following integrals:
$\int\frac{1}{\sin^3\text{x}\cos\text{x}}\text{dx}$
→
Evaluate the following integrals:
$\int\limits^{\frac{3}{2}}_0\big|\text{x}\sin\pi\text{x}\big|\text{dx}$
→
Find the angles of a triangle whose vertices are A (0, -1 ,-2), B(3, 1 ,4) and C(5 ,7 ,1).
→
Find the particular solution of the following differential equation;
$\frac{\text{dx}}{\text{dy}} = 1 + x^2 + y^2 + x^2y^2, $ given that $y = 1$ when $x = 0$.
→
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is $\pi:(\pi+2)$.
→
Integrate the function in Exercise:
$\text{e}^\text{x}\Bigg(\frac{1+\sin\text{x}}{1+\cos\text{x}}\Bigg)$
→
For the matrix $A = \left[ {\begin{array}{*{20}{c}} 1&1&1 \\ 1&2&{ - 3} \\ 2&{ - 1}&3 \end{array}} \right]$, show that $A^3 - 6A^2 + 5A + 11I = 0.$ Hence find $A^{-1}.$
→
By using properties of determinants, prove the following:$\begin{vmatrix} \text {x + 4} & \text{2x} & \text{2x} \\ \text{ 2x} & \text{x + 4} & \text{2x} \\ \text{2x} & \text{2x} & \text{x + 4} \end{vmatrix} = ( 5\text{x} + 4) (4 -\text{x}) = 1. $
→
The probabilities of two students A and B coming to the school in time are $\frac{3}{7}$ and $\frac{5}{7}$ respectively. Assuming that the events, 'A coming in time' and 'B coming in time' are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.
→