Purpose: Self-check on limits, continuity, and the derivative from first principles. Not graded — retake as needed.
Score: 0 / 12
Question 1
Evaluate \( \displaystyle \lim_{x \to 3}\, (2x^2 - 5x + 1) \).
Solution:
Direct substitution: \(2(9)-5(3)+1=18-15+1=4\).
Question 2
Evaluate \( \displaystyle \lim_{x\to 2}\frac{x^2-4}{x-2} \).
Solution:
Factor: \(\frac{(x-2)(x+2)}{x-2}=x+2\). As \(x\to 2\), limit \(=4\).
Question 3
Evaluate \( \displaystyle \lim_{x\to 0}\frac{\sqrt{x+9}-3}{x} \) (decimal).
Solution:
Rationalize numerator: \(\frac{(x+9)-9}{x(\sqrt{x+9}+3)}=\frac{1}{\sqrt{x+9}+3}\to \frac{1}{6}\approx 0.1667\).
Question 4
For \( f(x)=\begin{cases}x+1, & x<2\\ x^2-1, & x\ge 2\end{cases}\), is \(f\) continuous at \(x=2\)?
Solution:
Left: \(2+1=3\). Right: \(4-1=3\). \(f(2)=3\). All agree → continuous.
Question 5
Find \(k\) so \( g(x)=\begin{cases} kx-1, & x\le 1\\ x^2+2, & x>1\end{cases}\) is continuous at \(x=1\).
Solution:
Set \(k(1)-1=1+2=3\) → \(k=4\).
Question 6
Use first principles for \(f(x)=x^2-3x\). Find \(f'(2)\).
Solution:
\(f'(x)=\lim_{h\to 0}\frac{(x+h)^2-3(x+h)-(x^2-3x)}{h}=\lim_{h\to 0}(2x+h-3)=2x-3\). At 2: \(1\).
Question 7
Slope of tangent to \(y=\sqrt{x}\) at \(x=4\):
Solution:
\(f'(x)=\frac{1}{2\sqrt{x}}\); at 4: \(\frac{1}{4}\).
Question 8
Find \(\displaystyle \lim_{h\to 0}\frac{(3+h)^2-9}{h}\).
Solution:
\(\frac{6h+h^2}{h}=6+h\to 6\). (This is \(f'(3)\) for \(f=x^2\).)
Question 9
A ball's height: \(h(t)=-5t^2+20t\) (m, s). Average velocity on \([1,3]\)?
Solution:
\(h(3)=15,\ h(1)=15\). \((15-15)/2=0\) m/s.
Question 10
Same \(h(t)\): instantaneous velocity at \(t=1\)?
Solution:
\(h'(t)=-10t+20\). At 1: \(10\) m/s.
Question 11
\( \displaystyle \lim_{x\to\infty}\frac{3x^2-5}{2x^2+x-1} = \)
Solution:
Same degree → ratio of leading coefficients: \(3/2\).
Question 12
True/False: If \(\lim_{x\to a}f(x)\) exists, then \(f\) must be continuous at \(a\).
Solution:
False — also need \(f(a)\) defined and equal to the limit (e.g. removable hole).
📊 Self-Reflection
Rate confidence (1=need help, 4=mastered):
Topics rated 1 or 2: Re-watch lesson videos and retake those questions.