# [[Limit of the Square Root Function]] (Mathematics > Calculus) ## Definition The limit of the square root function at a point $c$ is equal to the square root of that point, provided $c > 0$. However, the limit does not exist for $c \leq 0$. The general limit of $\sqrt{x}$ as $x$ approaches $c$ is defined as: $ \lim_{x \to c} \sqrt{x} = \begin{cases} \sqrt{c} & \text{if } c > 0 \\ \text{DNE (does not exist)} & \text{if } c \leq 0 \end{cases} $ ## Key Concepts - The square root function, $f(x) = \sqrt{x}$, is only defined for $x \geq 0$. - Limits involving $\sqrt{x}$ for $x \to c$ depend on the value of $c$. - Limits from the left (i.e., as $x \to c^-$) are undefined for $c \leq 0$. ## Important Properties 1. $\lim_{x \to 4} \sqrt{x} = 2$ since $\sqrt{4} = 2$. 2. For $c < 0$, $\lim_{x \to c} \sqrt{x}$ does not exist because $\sqrt{x}$ is not defined for negative values. 3. $\lim_{x \to 0^+} \sqrt{x} = 0$, but $\lim_{x \to 0^-} \sqrt{x}$ does not exist because $\sqrt{x}$ is undefined for $x < 0$. ## Essential Formulas - $\lim_{x \to 4} \sqrt{x} = \sqrt{4} = 2$ - $\lim_{x \to 0} \sqrt{x} = 0$ (from the right side only) ## Core Examples 1. **Basic Example**: $ \lim_{x \to 9} \sqrt{x} = \sqrt{9} = 3 $ 2. **Advanced Example**: For $x \to 0^+$, $\lim_{x \to 0^+} \sqrt{x} = 0$, but $\lim_{x \to 0^-} \sqrt{x}$ does not exist because the square root function is not defined for negative $x$. ## Related Theorems/Rules - [[Continuity of Square Root Function]]: The square root function is continuous and differentiable for $x > 0$ but not defined for $x < 0$. - [[One-Sided Limits]]: Limits involving the square root function must consider direction (left or right). ## Common Pitfalls - Misinterpreting the square root function as defined for negative inputs. - Assuming the limit at $x = 0$ exists from both directions. ## Related Topics - [[Limits]] - [[One-Sided Limits]] - [[Continuity]] - [[Power Rule of Limits]] - [[Domain of the Square Root Function]] - [[Limits]] - [[Continuity]] ## Quick Review Questions 1. Why does the limit $\lim_{x \to 0} \sqrt{x}$ not exist for $x \to 0^-$? 2. What is the value of $\lim_{x \to 16} \sqrt{x}$? *** ![[Limit_of_the_Square_Root_Function_visualization]]