Question
Show that $\text{f(x)}=|\cos\text{x}|$ is a continuous function.
✓
Answer
If f is a real function on a subset of the real numbers and c be a point in the domain off, then f is
continuous at c is $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{f(x)}=\text{f(c)}$
Step I:
Let $\text{g(x)}=|\text{x}|$
$\text{h(x)}=\cos\text{x}$
$\text{f(x)}=(\text{goh})(\text{x})$
$=\text{g}(\text{h(x)})$
$=\text{g}(\cos\text{x})$
$={|\cos\text{x}|}$
$\text{g(x)}=|\text{x}|$ and $\text{h(x)}=\cos\text{x}$
Both are continuous for all values of $\text{x}\in\text{R}$
Step II:
(goh)(x) is also continuous
$\text{f(x)}=(\text{goh})(\text{x})$
$={|\cos\text{x}|}$
$\text{f(x)}={|\cos\text{x}|}$ is continuous for all values of $\text{x}\in\text{R}$
continuous at c is $\lim\limits_{{\text{x}}\rightarrow\text{c}}\text{f(x)}=\text{f(c)}$
Step I:
Let $\text{g(x)}=|\text{x}|$
$\text{h(x)}=\cos\text{x}$
$\text{f(x)}=(\text{goh})(\text{x})$
$=\text{g}(\text{h(x)})$
$=\text{g}(\cos\text{x})$
$={|\cos\text{x}|}$
$\text{g(x)}=|\text{x}|$ and $\text{h(x)}=\cos\text{x}$
Both are continuous for all values of $\text{x}\in\text{R}$
Step II:
(goh)(x) is also continuous
$\text{f(x)}=(\text{goh})(\text{x})$
$={|\cos\text{x}|}$
$\text{f(x)}={|\cos\text{x}|}$ is continuous for all values of $\text{x}\in\text{R}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Maximum Z = x - 5y + 20
Subject to
$\text{x}-\text{y}\geq0$
$-\text{x}+2\text{y}\geq2$
$\text{x}\geq3$
$\text{y}\geq4$
$\text{x},\text{y}\geq0$
→Subject to
$\text{x}-\text{y}\geq0$
$-\text{x}+2\text{y}\geq2$
$\text{x}\geq3$
$\text{y}\geq4$
$\text{x},\text{y}\geq0$
Evaluate the following intregals:
$\int\frac{\text{x}}{\sqrt{\text{x}^2+6\text{x}+10}}\ \text{dx}$
→$\int\frac{\text{x}}{\sqrt{\text{x}^2+6\text{x}+10}}\ \text{dx}$
Evaluate the following integrals:$\int\limits^{\pi}_0\text{x}\sin^3\text{x}\text{ dx}$
→Two sides of a parallelogram are $3 \hat{i}+4 \hat{j}-5 \hat{k}$ and $-2 \hat{j}+7 \hat{k}$. Find the unit vectors parallel to the diagonals.
→If the points with position vectors $10\hat{\text{i}}+3\hat{\text{j}},\ 12\hat{\text{i}}-5\hat{\text{j}}$ and $\text{a}\hat{\text{i}}+11\hat{\text{j}}$ are collinear, find the value of a.
→If the lines $\frac{\text{x}-1}{-3}=\frac{\text{y}-2}{-2\text{k}}=\frac{\text{z}-3}{2}$ and $\frac{\text{x}-1}{\text{k}}=\frac{\text{y}-2}{1}=\frac{\text{z}-3}{5}$ are perpendicular, find the value of k and, hence find the equation of the plane containing these lines.
→Find the vector equation of the line passing through the point (1, -1, 2) and perpendicular to the plane 2x - y + 3z - 5 = 0.
A factory produced two types of chemicals A and B. The following table gives the units of ingredients P and Q (per kg) of chemicals A and B as well as minimum requirements of P and Q and also cost per kg. chemicals A and B :
Find the number of units of chemicals A and B should be produced so as to minimize the cost.
Find the area of the region $\Bigg\{(\text{x},\text{y}): \frac{\text{x}^{2}}{\text{a}^{2}}+\frac{\text{y}^{2}}{\text{b}^{2}}<1< \frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}\bigg\}$
→Find the distance of the point (3, 3, 3) from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$
→