Limits and Continuity Worksheet
A Grade 12 Calculus worksheet covering limits, continuity, and related concepts with various question types.
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Limits and Continuity Worksheet
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Read each question carefully and provide your answer in the space provided. Show all your work for full credit.
1. What is the value of \(\lim_{x \to 2} (3x^2 - 5x + 1)\)?
1
3
5
7
2. For a function to be continuous at a point c, which of the following conditions must be met?
\(f(c)\) is defined
\(\lim_{x \to c} f(x)\) exists
\(\lim_{x \to c} f(x) = f(c)\)
All of the above
3. A function is said to be at a point if the limit of the function as x approaches that point exists, the function is defined at that point, and the limit equals the function's value at that point.
4. The limit of \(f(x) = \frac{x^2 - 4}{x - 2}\) as x approaches 2 is .
5. Evaluate the limit: \(\lim_{x \to 0} \frac{\sin(x)}{x}\).
6. Determine if the function \(f(x) = \begin{cases} x^2+1, & x < 1 \cr 3x-1, & x \ge 1 \end{cases}\) is continuous at \(x=1\). Show your work.
7. If a function is differentiable at a point, then it must be continuous at that point.
True
False
8. A removable discontinuity occurs when the left-hand limit and the right-hand limit are equal but not equal to the function's value at that point, or the function is undefined at that point.
True
False
9. Consider the graph of a function f(x) below. Based on the graph, answer the following questions:
a) \(\lim_{x \to 2^-} f(x) = \)
b) \(\lim_{x \to 2^+} f(x) = \)
c) Is f(x) continuous at \(x=2\)? Explain.