Math Problem Statement
h. lim π₯β 1 π(π₯)ππ π (π₯) = { π₯ + 3 ππ π₯ < 1 2 π₯ ππ π₯ = 1 β 5 π₯ β 1 ππ π₯ > 1 (Provide a tables of values)
Solution
To find the limit of as approaches 1, we'll need to evaluate the left-hand limit () and the right-hand limit (). Here's the piecewise function defined:
\begin{cases} x + 3 & \text{if } x < 1 \\ 2x & \text{if } x = 1 \\ \sqrt{5x - 1} & \text{if } x > 1 \end{cases}$$ ### Left-hand Limit ($$x \to 1^-$$) For $$x < 1$$, $$f(x) = x + 3$$. As $$x$$ approaches 1 from the left: $$\lim_{x \to 1^-} f(x) = 1 + 3 = 4$$ ### Right-hand Limit ($$x \to 1^+$$) For $$x > 1$$, $$f(x) = \sqrt{5x - 1}$$. As $$x$$ approaches 1 from the right: $$\lim_{x \to 1^+} f(x) = \sqrt{5(1) - 1} = \sqrt{4} = 2$$ ### Table of Values Around $$x = 1$$ Let's evaluate $$f(x)$$ for values around 1 to observe the behavior: | $$x$$ | $$f(x)$$ if $$x < 1$$ | $$f(x)$$ at $$x = 1$$ | $$f(x)$$ if $$x > 1$$ | |----------|----------------------------|---------------------------|---------------------------| | 0.9 | $$0.9 + 3 = 3.9$$ | - | - | | 0.99 | $$0.99 + 3 = 3.99$$ | - | - | | 0.999 | $$0.999 + 3 = 3.999$$ | - | - | | 1 | - | $$2(1) = 2$$ | - | | 1.001 | - | - | $$\sqrt{5(1.001) - 1} \approx 2.002$$ | | 1.01 | - | - | $$\sqrt{5(1.01) - 1} \approx 2.02$$ | | 1.1 | - | - | $$\sqrt{5(1.1) - 1} \approx 2.19$$ | ### Conclusion - The left-hand limit as $$x$$ approaches 1 is 4. - The right-hand limit as $$x$$ approaches 1 is 2. - Since the left-hand limit and the right-hand limit are not equal, the limit of $$f(x)$$ as $$x$$ approaches 1 does not exist. Would you like more details or have any questions? --- **5 Related Questions:** 1. What is the importance of checking both left-hand and right-hand limits? 2. How do you determine the continuity of a piecewise function at a point? 3. Can the limit exist if the left-hand and right-hand limits are different? 4. How do you compute the limit when one side approaches infinity? 5. What are some techniques to find the limit of more complex piecewise functions? **Tip:** Always check both left-hand and right-hand limits to determine if the overall limit exists at a point.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
Limit of a function as x approaches a value
Square root function
Piecewise definition of functions
Theorems
Left-hand and Right-hand Limit Theorem
Continuity Definition
Suitable Grade Level
Grades 10-12