Find lim as x approaches -3 of h(x).
Understand the Problem
The question is asking to find the limit of the piecewise function h(x) as x approaches -3. It provides different expressions for h(x) based on the value of x, and the goal is to determine the limit when x approaches -3.
Answer
$$ -5 $$
Answer for screen readers
$$ -5 $$
Steps to Solve
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Identify the piece of the function relevant to the limit To find $\lim_{x \to -3} h(x)$, we notice that $-3 < -2$. Therefore, we use the piece of the function defined for $x < -2$: $$ h(x) = 2x + 1 $$
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Substitute the value into the function Now, we substitute $x = -3$ into the expression $2x + 1$: $$ h(-3) = 2(-3) + 1 $$
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Calculate the value Calculating the above expression gives: $$ h(-3) = -6 + 1 = -5 $$
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Conclusion on the limit Since the piece of the function is continuous for $x < -2$, we conclude that: $$ \lim_{x \to -3} h(x) = -5 $$
$$ -5 $$
More Information
The limit represents the value that the function approaches as $x$ approaches -3. Since we used the valid piece of the piecewise function for values less than -2, we confirmed that the limit exists and is equal to -5.
Tips
- Forgetting the correct piece of the function: Always ensure you select the right piece of the piecewise function for the specific limit you are evaluating.
- Not substituting correctly: Double-check your arithmetic when substituting values into the function.
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