\(\textbf{1)}\) Find \( \displaystyle \lim_{x\to 2^{-}} f(x) \)
where \(f(x) = \begin{cases}
5x+3 & \text{if } x \lt 2 \\
4x & \text{if } x \geq 2
\end{cases}\)
\(\textbf{2)}\) Find \( \displaystyle \lim_{x\to 2^{+}} f(x) \)
where \(f(x) = \begin{cases}
5x+3 & \text{if } x \lt 2 \\
4x & \text{if } x \geq 2
\end{cases}\)
\(\textbf{3)}\) Find \( \displaystyle \lim_{x\to 2} f(x) \)
where \(f(x) = \begin{cases}
5x+3 & \text{if } x \lt 2 \\
4x & \text{if } x \geq 2
\end{cases}\)
\(\textbf{4)}\) Find \( f(2) \)
where \(f(x) = \begin{cases}
5x+3 & \text{if } x \lt 2 \\
4x & \text{if } x \geq 2
\end{cases}\)
\(\textbf{5)}\) Find \( \displaystyle \lim_{x\to 4^{-}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{6)}\) Find \( \displaystyle \lim_{x\to 4^{+}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{7)}\) Find \( \displaystyle \lim_{x\to 6^{-}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{8)}\) Find \( \displaystyle \lim_{x\to 6^{+}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{9)}\) Is the following continuous at \(x=4? \)
\(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{10)}\) Is the following continuous at \( x=6? \)
\(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{11)}\) Find the value of a that makes the function continuous.
\(f(x) = \begin{cases}
\frac{x^2+6x+5}{x+1} & \text{if } x \neq -1 \\
a & \text{if }x = -1
\end{cases}\)
\(\textbf{12)}\) Find the values of a and b that make the function continuous at all points.
\(f(x) = \begin{cases}
2x^2 & \text{if } x\leq 2 \\
ax+b & \text{if } 2\lt x \lt 4 \\
x^2+4 & \text{if }x\geq 4
\end{cases}\)
\(\textbf{13)}\) Find the value of k that makes the function continuous at all points.
\(f(x) = \begin{cases}
\sin{x} & \text{if } x\leq \pi \\
x-k & \text{if } x\geq \pi
\end{cases}\)
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Limits on Graphs}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Continuity on Graphs}\)
\(\,\,\,\,\,\,\,\,\)\(…\)
\(\bullet\text{ Infinite Limits}\)
\(\,\,\,\,\,\,\,\,\displaystyle \lim_{x\to 4^{+}} \frac{5}{x-4}\)
\(\bullet\text{ Limits at Infinity}\)
\(\,\,\,\,\,\,\,\,\displaystyle\lim_{x\to \infty}\frac{5x^2+2x-10}{3x^2+4x-5}\)
\(\bullet\text{ Trig Limits}\)
\(\,\,\,\,\,\,\,\,\displaystyle \lim_{\theta\to0} \frac{\sin \theta}{\theta}=1\)
In Summary
Piecewise functions can be helpful for modeling real-world situations where a function behaves differently over different intervals.
Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a particular value.
Continuity is another important concept in calculus that refers to the smoothness of a function. A function is considered continuous if it can be drawn without lifting the pen from the paper. Continuity is important for understanding how a function behaves and for making predictions about its values.
Piecewise functions, limits, and continuity are typically studied in advanced math courses, such as calculus or analysis. These concepts build upon foundational knowledge of algebra and trigonometry and are used to understand and analyze the behavior of functions.