Math Problem Statement
Let f(x) be defined as:
f(x) = { 3x^3 - 2m, x ≤ -1; 4x^2 + 5m, x > -1 }
(a) Find (in terms of m) lim x → -1^+ f(x)
(b) Find (in terms of m) lim x → -1^- f(x)
(c) Find the value of m so that lim x → -1^- f(x) = lim x → -1^+ f(x)
Solution
Let's work through the problem step by step.
The function is defined as:
3x^3 - 2m, & x \leq -1 \\ 4x^2 + 5m, & x > -1 \end{cases}$$ #### Part (a) - Find $$\lim_{x \to -1^+} f(x)$$ in terms of $$m$$: This limit represents the behavior of the function as $$x$$ approaches $$-1$$ from the right (i.e., for values greater than $$-1$$). For $$x > -1$$, $$f(x) = 4x^2 + 5m$$. So, we evaluate the limit of this expression as $$x \to -1^+$$: $$\lim_{x \to -1^+} f(x) = 4(-1)^2 + 5m = 4 + 5m$$ #### Part (b) - Find $$\lim_{x \to -1^-} f(x)$$ in terms of $$m$$: This limit represents the behavior of the function as $$x$$ approaches $$-1$$ from the left (i.e., for values less than or equal to $$-1$$). For $$x \leq -1$$, $$f(x) = 3x^3 - 2m$$. So, we evaluate the limit of this expression as $$x \to -1^-$$: $$\lim_{x \to -1^-} f(x) = 3(-1)^3 - 2m = -3 - 2m$$ #### Part (c) - Find the value of $$m$$ such that: $$\lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x)$$ From parts (a) and (b), we know that: $$-3 - 2m = 4 + 5m$$ Solving this equation for $$m$$: $$-3 - 2m = 4 + 5m$$ Add $$2m$$ to both sides: $$-3 = 4 + 7m$$ Subtract $$4$$ from both sides: $$-7 = 7m$$ Divide by $$7$$: $$m = -1$$ ### Summary of Results: - (a) $$\lim_{x \to -1^+} f(x) = 4 + 5m$$ - (b) $$\lim_{x \to -1^-} f(x) = -3 - 2m$$ - (c) The value of $$m$$ that makes the limits equal is $$m = -1$$. Would you like further details or explanations on any part of the solution? ### Related Questions: 1. How would the limits change if the function were defined differently for $$x \leq -1$$? 2. What happens to the function's continuity if $$m \neq -1$$? 3. Can you prove the function is continuous at $$x = -1$$ for the found value of $$m$$? 4. How do we calculate limits for piecewise functions in general? 5. What is the geometric interpretation of the limits in this context? ### Tip: Always check both left-hand and right-hand limits to ensure a function is continuous at a point, especially for piecewise functions.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
lim x → -1^+ f(x) = 4 + 5m
lim x → -1^- f(x) = -3 - 2m
Theorems
Limit definition and continuity for piecewise functions
Suitable Grade Level
Grade 12 or college-level calculus