Math Problem Statement
Evaluate the limits. f(x)={|6x|x0x≠0x=0 Enter DNE if the limit does not exist. a) limx→0−f(x) =
b) limx→0+f(x) =
c) limx→0f(x) =
Solution
We are asked to evaluate the left-hand limit, right-hand limit, and two-sided limit of the piecewise function defined as:
\begin{cases} |6x| & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$$ Let's break down each limit. ### a) $$\lim_{x \to 0^-} f(x)$$ For $$x \to 0^-$$ (approaching zero from the left), $$x$$ is negative. The absolute value of $$6x$$ is: $$|6x| = -6x \quad \text{(since $$x$$ is negative)}$$ Thus, for $$x < 0$$, $$f(x) = -6x$$. Now, taking the limit as $$x \to 0^-$$: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-6x) = -6(0) = 0$$ So, $$\lim_{x \to 0^-} f(x) = 0$$ ### b) $$\lim_{x \to 0^+} f(x)$$ For $$x \to 0^+$$ (approaching zero from the right), $$x$$ is positive. The absolute value of $$6x$$ is: $$|6x| = 6x \quad \text{(since $$x$$ is positive)}$$ Thus, for $$x > 0$$, $$f(x) = 6x$$. Now, taking the limit as $$x \to 0^+$$: $$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} 6x = 6(0) = 0$$ So, $$\lim_{x \to 0^+} f(x) = 0$$ ### c) $$\lim_{x \to 0} f(x)$$ For the two-sided limit to exist, both the left-hand and right-hand limits must be equal. Since: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = 0$$ we can conclude that: $$\lim_{x \to 0} f(x) = 0$$ ### Final answers: a) $$\lim_{x \to 0^-} f(x) = 0$$ b) $$\lim_{x \to 0^+} f(x) = 0$$ c) $$\lim_{x \to 0} f(x) = 0$$ Would you like more details or have any questions? Here are some related questions for further practice: 1. What is the definition of a one-sided limit? 2. How do you evaluate limits involving absolute values? 3. What are the conditions for the existence of a two-sided limit? 4. What happens if the left-hand and right-hand limits are different? 5. How does the piecewise definition of a function affect its limits? **Tip:** When evaluating limits involving absolute values, always break them down based on whether the input is positive or negative.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Absolute Value
One-sided Limits
Two-sided Limits
Formulas
lim x→c f(x)
|x| = x if x ≥ 0, |x| = -x if x < 0
Theorems
Limit Laws
Properties of Absolute Value
Existence of Two-sided Limits
Suitable Grade Level
Grades 11-12, Early College