Property 15
Limit of Piecewise Function
\(a\in\mathbb R\)
\(f(x)=\begin{cases}g(x)&x\ge a\\h(x)&x\lt a\end{cases}\)
a point is the critical point of f function. On critical points right-hand and left-hand limits are to be analysed.
\(\lim\limits_{x\to a^+}f(x)=g(a)\)
\(\lim\limits_{x\to a^-}f(x)=h(a)\)
1.
\(f(x)=\begin{cases}2x+1,&x\ge1\\x-2,&x<1\end{cases}\)
\(\Rightarrow \lim\limits_{x\to1^-}f(x)=?\)
2.
\(f(x)=\begin{cases}x-3,&x>2\\3x+1,&x\le2\end{cases}\)
\(\Rightarrow \lim\limits_{x\to2^+}f(x)=?\)
3.
\(f(x)=\begin{cases}x^2-1,&x>2\\3x+2,&x\le2\end{cases}\)
\(\Rightarrow \lim\limits_{x\to2}f(x)=?\)
4.
\(f(x)=\begin{cases}3x+1,&x>1\\x^2+3,&x\le1\end{cases}\)
\(\Rightarrow \lim\limits_{x\to1}f(x)=?\)
5.
\(f(x)=\begin{cases}x^2+3,&x>-2\\5-x,&x\le-2\end{cases}\)
\(\Rightarrow \lim\limits_{x\to-2}f(x)=?\)
6.
\(a,k\in\mathbb R\)
\(f(x)=\begin{cases}kx+4,&x>-1\\-2x^3,&x\le-1\end{cases}\)
\(\lim\limits_{x\to-1}f(x)=a\)
\(\Rightarrow k=?\)
7.
\(a,k\in\mathbb R\)
\(f(x)=\begin{cases}kx+2,&x>1\\x+4,&x\le1\end{cases}\)
\(\lim\limits_{x\to1}f(x)=a\)
\(\Rightarrow k=?\)
8.
\(f(x)=\begin{cases}2x-1,&x>2\\a,&x=2\\x+1,&x<2\end{cases}\)
\(\lim\limits_{x\to2}f(x)=f(2)\)
\(\Rightarrow a=?\)
9.
\(f(x)=\begin{cases}2x,&x>1\\a,&x=1\\x^2+1,&x<1\end{cases}\)
\(\lim\limits_{x\to1}f(x)=f(1)\)
\(\Rightarrow a=?\)
10.
\(f(x)=\begin{cases}3x+2,&x>1\\kx-1,&x\le1\end{cases}\)
\(\lim\limits_{x\to1}f(x)=f(1)\)
\(\Rightarrow k=?\)
11.
\(f(x)=\begin{cases}2x+b,&x>1\\6,&x=1\\ax+1,&x<1\end{cases}\)
\(\lim\limits_{x\to1}f(x)=f(1)\)
\(\Rightarrow a+b=?\)
12.
\(f(x)=\begin{cases}x^2+a,&x<-1\\4,&x=-1\\bx+1,&x>-1\end{cases}\)
\(\lim\limits_{x\to-1}f(x)=f(-1)\)
\(\Rightarrow a\cdot b=?\)
13.
\(f(x)=\begin{cases}x-4a,&x>1\\5,&x=1\\x^2+b,&x<1\end{cases}\)
\(\lim\limits_{x\to1}f(x)=f(1)\)
\(\Rightarrow a+b=?\)