Solution: Provided function is $f(x)=|x|-|x+1|$ When the value of $x<-1: f(x)=-x-\{-(x+1)\}=-x+x+1=1$ When the value of $-1 \leq x<0 ; f(x)=-x-(x+1)=-2 x-1$ When the value of $x \geq 0, ;...

Solution: Let's assume $f(x)=|x|$ and $g(x)=\sin x$, therefore $($ gof $) x=g\{f(x)\}=g(|x|)=\sin |x|$ So now, $f$ and $g$ are continuous, as a result their composite, (gof) is also continuous. As a...

Solution: The provided function is $f(x)=|\cos x|$ $\mathrm{f}(\mathrm{x})$ is real and finite for all $x \in R$ and Domain of $f(x)$ is $R$. Let's assume that $g(x)=\cos x$ and $h(x)=|x|$ Provided...

Solution: The provided function is : $f(x)=\cos \left(x^{2}\right)$ Let's assume $g(x)=\cos x$ and $h(x)=x^{2}$, therefore $\operatorname{goh}(x)=g(h(x))$ $=g\left(x^{2}\right)$ $=\cos...

Solution: The provided function is: $f(x)=\left\{\begin{array}{cl}5, & \text { if } & x \leq 2 \\ a x+b, & \text { if } & 2<x<10 \\ 21, & \text { if } & x \geq...

Solution: The provided function is $f(x)= \begin{cases}k x+1, & \text { if } x \leq 5 \\ 3 x-5, & \text { if } x>5\end{cases}$ When the value of $x<5, f(x)=k x+1$ : A polynomial is...

Solution: The provided function is: $f(x)=\left\{\begin{array}{lll}k x+1, \text { if } x \leq \pi \\ \cos x, \text { if } x>\pi\end{array}\right.$ $\lim _{x \rightarrow n^{+}} f(x)=\lim _{h...

Solution: The provided function is $f(x)= \begin{cases}k x^{2}, & \text { if } x \leq 2 \\ 3, & \text { if } x>2\end{cases}$ $\lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0}...

Solution: The provided function is $f(x)=\left\{\begin{array}{ccc}\frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\ 3, & \text { if } x=\frac{\pi}{2}\end{array}\right.$ $\lim...

Solution: The provided function is $f(x)= \begin{cases}\sin x-\cos x, & \text { if } \quad x \neq 0 \\ -1, & \text { if } \quad x=0\end{cases}$ Let us now find the L.H.L and R.H.L. at $x=0$....

Solution: The provided function is: $f(x)=\left\{\begin{array}{cc}x^{2} \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$ $\lim _{x \rightarrow 0}...

Solution: The provided function is $f(x)=\left\{\begin{array}{lll}\frac{\sin x}{x}, & \text { if } & x<0 \\ x+1, & \text { if } & x \geq 0\end{array}\right.$ At $x=0$ Left Hand...

Solution: Continuity of cosine: Let's say that an arbitrary real number be "a" then, That implies, $\lim _{h \rightarrow 0}(\cos a \cos h-\sin a \sin h)$ $=\cos a \lim _{h \rightarrow 0} \cos h-\sin...

Solution: (a) Let an arbitrary real number be "a" therefore, $\lim _{x \rightarrow a^{-}} f(x) \Rightarrow \lim _{h \rightarrow 0} f(a+h)$ So now, $\lim _{h \rightarrow 0} f(a+h)=\lim _{h...

Solution: (a) Let's assume that any arbitrary real number is "a" then $\lim f(x) \Rightarrow \lim f(a+h)$ So now, $\lim _{h \rightarrow 0}(\sin a \cos h+\cos a \sin h+\cos a \cos h-\sin a \sin h)$...

Solution: The provided function is $f(x)=x^{2}-\sin x+5$ Left Hand Limit $=\lim _{x \rightarrow z^{-}}\left(x^{2}-\sin x+5\right)=\lim _{x \rightarrow n^{-}}\left[(\pi-h)^{2}-\sin...

Solution: Here, for any real number, $x$, The fractional part or decimal part of $x$ is denoted by $[x]$. For example let us consider, $[2.35]=0.35$ $[2]=0$ $[-5]=0$ The function $g: R->R$...

Solution: As at $x=0$, $f(x)$ is continuous . Therefore, Left Hand Limit =$\lim _{x \rightarrow 0^{-}} f(x)=f(0)=\lambda\left(x^{2}-2 x\right)=\lambda(0-0)=0$ And Right Hand Limit = $\lim _{x...

Solution: The function provided is $f(x)=\left\{\begin{array}{lll}a x+1, & \text { if } & x \leq 3 \\ b x+3, & \text { if } & x>3\end{array}\right.$ Check Continuity at $x=3$,...

Solution: The provided function is $f(x)=\left\{\begin{array}{lll}-2, & \text { if } & x \leq-1 \\ 2 x, & \text { if } & -1<x \leq 1 \\ 2, & \text { if } &...

Solution: The function provided is $f(x)=\left\{\begin{array}{lll}2 x, & \text { if } & x<0 \\ 0, & \text { if } & 0 \leq x \leq 1 \\ 4 x, & \text { if } &...

Solution: The provided function is $f(x)= \begin{cases}3, & \text { if } 0 \leq x \leq 1 \\ 4, & \text { if } 1<x<3 \\ 5, & \text { if } 3 \leq x \leq 10\end{cases}$ In interval,...

Solution: The function provided is $f(x)= \begin{cases}x+5, & \text { if } \quad x \leq 1 \\ x-5, & \text { if } \quad x>1\end{cases}$ At $x=1$, Left Hand Limit = $\lim _{x \rightarrow...

Solution: The function provided is $f(x)=\left\{\begin{array}{ccc}x^{10}-1, & \text { if } & x \leq 1 \\ x^{2}, & \text { if } & x>1\end{array}\right.$ At $x=1$, Left Hand Limit...

Solution: The provided function is $f(x)=\left\{\begin{array}{lll}x^{3}-3, & \text { if } & x \leq 2 \\ x^{2}+1, & \text { if } & x>2\end{array}\right.$ At $x=2$, Left Hand Limit...

Solution: The function provided is $f(x)=\left\{\begin{array}{lll}\frac{x}{|x|}, & \text { if } & x<0 \\ -1, & \text { if } & x \geq 0\end{array}\right.$ At $x=0$, Left Hand Limit...

Solution: The function provided is $f(x)= \begin{cases}\frac{|x|}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}$ Also the $\mathrm{f}(\mathrm{x})=|\mathrm{x}| / \mathrm{x}$...

Solution: The provided function is $f(x)= \begin{cases}|x|+3, & \text { if } x \leq-3 \\ -2 x, & \text { if }-3<x<3 \\ 6 x+2, & \text { if } x \geq 3\end{cases}$ Given here the...

Solution: The function provided is $f(x)=\left\{\begin{array}{lll}2 x+3, & \text { if } & x \leq 2 \\ 2 x-3, & \text { if } & x>2\end{array}\right.$ Given here the $f(x)$ is...

Solution: The provided function is $f(x)=x^{n}$ in which $n$ is a positive integer. Continuity at $x=n, \lim _{x \rightarrow n} f(x)=\lim _{x \rightarrow n}\left(x^{n}\right)=n^{n}$ And $f(n)=n^{n}$...