\(\textbf{1)}\) 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 \(f(x) = \begin{cases}
5x+3 & \text{if } x \lt 2 \\
4x & \text{if } x \geq 2
\end{cases}\)
\(\textbf{3)}\) \( \displaystyle \lim_{x\to 2^{+}} \frac{|x-2|}{x-2} \)
\(\textbf{4)}\) \( \displaystyle \lim_{x\to 2^{-}} \frac{|x-2|}{x-2} \)
\(\textbf{5)}\) \( \displaystyle \lim_{x\to 2} \frac{|x-2|}{x-2} \)
\(\textbf{6)}\) \( \displaystyle \lim_{x\to 1^{+}} \frac{3|x-1|}{x-1} \)
\(\textbf{7)}\) \( \displaystyle \lim_{x\to 1^{-}} \frac{3|x-1|}{x-1} \)
\(\textbf{8)}\) \( \displaystyle \lim_{x\to 1} \frac{3|x-1|}{x-1} \)
\(\textbf{9)}\) Find \( \displaystyle \lim_{x\to 4^{-}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{10)}\) Find \( \displaystyle \lim_{x\to 4^{+}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{11)}\) Find \( \displaystyle \lim_{x\to 6^{-}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{12)}\) Find \( \displaystyle \lim_{x\to 6^{+}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{13)}\) Find \( \displaystyle \lim_{x\to 7^{-}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{14)}\) Find \( \displaystyle \lim_{x\to 7^{+}} f(x) \)
where \(f(x) = \begin{cases}
-x+5 & \text{if } x\leq 4 \\
x-3 & \text{if } 4\lt x \lt 6 \\
x & \text{if }x\geq 6
\end{cases}\)
\(\textbf{15)}\) \( \displaystyle \lim_{x\to 1^{-}} \frac{x^2+3x-4}{\left(x-1\right)^4} \)
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Limits on Graphs}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Continuity on Graphs}\)
\(\,\,\,\,\,\,\,\,\)\(…\)
\(\bullet\text{ Piecewise Functions- Limits and Continuity}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Limits at Infinity}\)
\(\,\,\,\,\,\,\,\,\displaystyle\lim_{x\to \infty}\frac{5x^2+2x-10}{3x^2+4x-5}…\)
\(\bullet\text{ Trig Limits}\)
\(\,\,\,\,\,\,\,\,\displaystyle \lim_{\theta\to0} \frac{\sin \theta}{\theta}=1…\)
In Summary
One-sided limits are a type of mathematical concept defined as the limit of a function as it approaches a certain value from one direction, either from the left or the right. In order for a limit to exist at a point, both the left and right limits must be the same. One of the main reasons we learn about one-sided limits is because for a lot of functions, the left limit doesn’t always equal the right limit.
One-sided limits are typically introduced in calculus courses.
One-sided limits were first formally introduced by Augustin-Louis Cauchy in the early 19th century.