Name: _____________________________________________________________
Find the limit \[ \lim_{x\rightarrow3}\frac{x^2+2}{x-3} \]
Sketch the graph of \(f(x)=\frac{x^2+2}{x-3}\).
Using your sketch, trace your finger along the graph from \(x=1\) toward \(x=3\) and then from \(x=5\) toward \(x=3\).
What do you think the limit might be?
What happens if we try a plug-in approach for this limit?
Now, plug numbers into \(f(x)\), getting closer to \(x=3\) from both directions. Follow along using the table below. What do you think the limit might be?
| \(x\) | \(2.9\) | \(2.99\) | \(2.999\) | \(3.001\) | \(3.01\) | \(3.1\) |
|---|---|---|---|---|---|---|
| \(\frac{x^2+2}{x-3}\) |
Consider \[ \lim_{x\rightarrow 0}\frac{\sqrt{x-1}-1}{x} \] Since we cannot divide by zero, we need to rewrite this. Since we have a square root in the numerator, one option is to rationalize it.
Rationalize the numerator of \(\frac{\sqrt{x-1}-1}{x}\).
Check your course notes for the replacement theorem. Does that apply here? Discuss with your group and jot down a brief explanation here.
Use the replacement theorem, along with the operation with limits for quotients to find \(\lim_{x\rightarrow 0}\frac{\sqrt{x-1}-1}{x}\).
The recommended dosage of a children’s pain reliever is 40mg for infants 0 to 3 months; 80mg for more than 3 up to 12 months; and 120mg for more than 12 up through 24 months.
Think about how we might use a piecewise function to represent these dosage recommendations.
What are the appropriate values for \(A\), \(B\), and \(C\)? \[
f(x)=\begin{cases}
40 \quad 0 < x \le 3\\
80 \quad A < x \le B\\
C \quad 12 < x \le 24
\end{cases}
\]
Find \[
\lim_{x\rightarrow 12^-}f(x)
\] and \[
\lim_{x\rightarrow 12^+}f(x)
\] A sketch may be helpful but is not required. what can we conclude about \(\lim_{x\rightarrow 12}f(x)\)?
This activity was adapted from examples in Applied Calculus for the Life and Social Sciences (Larson) by Dr. Lauren Cappiello at CSU Sacramento.