Math Problem Statement
If lim x = -5^-, f(x)=?
If lim x = -5^-, f(x) = ?
Solution
The function is defined piecewise, and you are asking for the limit of as , which means we approach from the left-hand side.
According to the piecewise function:
17 & \text{if } x \leq -5 \\ -x + 12 & \text{if } -5 \leq x < 7 \\ 7 & \text{if } x = 7 \\ 12e^{x-7} & \text{if } x > 7 \end{cases}$$ For $$x \to -5^-$$, we are approaching from the left, so we use the case where $$x \leq -5$$. In this case, $$f(x) = 17$$ for all $$x \leq -5$$. Thus, $$\lim_{x \to -5^-} f(x) = 17.$$ Would you like further details or have any questions? Here are 5 related questions for further exploration: 1. What is the limit of $$f(x)$$ as $$x \to -5^+$$? 2. Is $$f(x)$$ continuous at $$x = -5$$? 3. What is the limit of $$f(x)$$ as $$x \to 7^-$$? 4. What is $$f'(x)$$ for $$-5 \leq x < 7$$? 5. What is the value of $$\lim_{x \to +\infty} f(x)$$? **Tip:** For piecewise functions, always pay attention to which interval your $$x$$-value belongs to when taking limits or evaluating the function at specific points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Continuity
Formulas
Limit definition of a piecewise function
Piecewise function intervals
Theorems
Limit from the left-hand side
Continuity at a point
Suitable Grade Level
High School - Calculus