Limits & Continuity

  1. Determine the value of each limit, or determine that the limit doesn’t exist.
    \(\displaystyle \lim\limits_{x \to -\infty} f(x) \)
    \(\displaystyle \lim\limits_{x \to -2} f(x) \)
    \(\displaystyle \lim\limits_{x \to -1^-} f(x) \)
    \(\displaystyle \lim\limits_{x \to -1^+} f(x) \)
    \(\displaystyle \lim\limits_{x \to -1} f(x) \)
    \(\displaystyle \lim\limits_{x \to 0} f(x) \)
    \(\displaystyle \lim\limits_{x \to 1^-} f(x) \)
    \(\displaystyle \lim\limits_{x \to 1^+} f(x) \)
    \(\displaystyle \lim\limits_{x \to 1} f(x) \)
    \(\displaystyle \lim\limits_{x \to 2} f(x) \)
    \(\displaystyle \lim\limits_{x \to \infty} f(x) \)

  2. Let \(g\) be the function defined piecewise by the following four formulas: \[ g(x) = \begin{cases} -x &\text{for } x \lt 1 \\7 &\text{for } x=1 \\x^2-2 &\text{for } 1 \lt x \leq 5 \\7x-5 &\text{for } 5 \lt x \end{cases} \] Evaluate the following. If a limit doesn’t exist, cross it out with confidence.

    \( \displaystyle \lim\limits_{x \to 1^-} g(x)\)
    \( \displaystyle \lim\limits_{x \to 1} g(x)\)
    \( \displaystyle g(1)\)
    \( \displaystyle \lim\limits_{x \to 5^-} g(x)\)
    \( \displaystyle \lim\limits_{x \to 5^+} g(x)\)
    \( \displaystyle \lim\limits_{x \to 5} g(x)\)

  3. Use a calculator to numerically approximate the value of each of these limits accurate to with ±one-thousandth.

    \(\displaystyle \lim\limits_{x \to \infty} \left(\frac{x}{x+1}\right)^x \)
    \(\displaystyle \lim\limits_{x \to 0} \; 4\arctan\bigl(x^x\bigr) \)
    \(\displaystyle \lim\limits_{x \to 0} \frac{\cos(x)-\cos(2x)}{x^2} \)
  4. Is it possible that there is function \(f\) such that \(\lim_{x \to 3} f(x) = 7\) but \(f(3) = 5\,?\) If so, sketch a graph of such a function \(f.\) if not, explain why not.
  5. Sketch the graph of a function \(f\) that satisfies all of the following conditions: \[ \lim\limits_{x \to -\infty} f(x) = 2 \quad \lim\limits_{x \to 3^-} f(x) = 5 \quad \lim\limits_{x \to 3^+} f(x) = -\infty \quad \lim\limits_{x \to \infty} f(x) = \infty \]
  6. James Stewart Let \(c\) be the speed of light. According to the theory of relativity, if a particle’s mass at rest is \(m_0\) then its mass when travelling at a velocity of \(v\) will be \[ m = \frac{m_0}{\sqrt{1-v^2/c^2}}\,. \] What happens to the mass of the particle as its velocity approaches \(c\,?\)
  7. If we know that \(\lim_{x \to 3} f(x) = 7\) and \(\lim_{x \to 3} g(x) = 2\,,\) what must the value of \(\lim_{x \to 3} 2f(x)-x\bigl(g(x)-1\bigr)\) be?
  8. What must the value of \(\lim_{x \to 1} f(x)\) be if we know that \[ \lim\limits_{x \to 1} \frac{2-f(x)}{3-x^2} = -3\,? \]
  9. If \(\lim_{x \to c} \big(f(x) + g(x)\big) = 11\) and \(\lim_{x \to c} \big(f(x) - g(x)\big) = 7,\) what must the value of \(\lim_{x \to c} \big(f(x)g(x)\big)\) be?
  10. There exists a single value of the coefficient \(b\) such that the limit \( \lim_{x \to 7} \tfrac{x^2+bx-14}{x-7} \) exists; what is it?
  11. Let \(g\) be the function defined piecewise in terms of some function \(f\) as follows: \[ g(x) = \begin{cases} -1 &\text{ for } x \lt -3 \\f(x) &\text{ for } -3 \leq x \lt 1 \\-x^2+5 &\text{ for } x \geq 1 \end{cases} \] Find an example of a linear function \(f\) such that \(g\) is continuous.
  12. For functions \(f\) and \(g,\) it may be that \(\lim_{x \to c}\big(f(x) + g(x)\big)\) exists even though neither \(\lim_{x \to c}f(x)\) nor \(\lim_{x \to c}g(x)\) exist. Find examples of such functions.
  13. For functions \(f\) and \(g,\) it may be that \(\lim_{x \to c}\big(f(x)g(x)\big)\) exists even though neither \(\lim_{x \to c}f(x)\) nor \(\lim_{x \to c}g(x)\) exist. Find examples of such functions.
  14. How can you very quickly argue that the polynomial \(x^{171}-7x^2-1\) has a root on the domain \([0,2]\) by invoking the Intermediate Value Theorem?
  15. A hiker is camped at the base of Mt Whitney. She departs basecamp at 7am, hikes the trail up the mountain, and arrives at the summit at 7pm. The next morning, after camping overnight at the summit, the hiker again departs at 7am, hikes the trail down the mountain, and arrives back at basecamp at 7pm. Recalling the Intermediate Value Theorem, convince yourself that there must be some location on that trail up Mt Whitney that the hiker passed at the exact same time on the way up and on the way down.
  16. James Stewart Does there exist a number that is exactly one more than its cube? The challenge is to confidently answer this question without calculating any such number.
  17. Patrick Snyder Demonstrate how to compute the following limit algebraically. \[ \lim_{x\ \to 0} \frac{\sqrt{1+\tan(x)}-\sqrt{1+\sin(x)}}{x^3} \]
  18. Suppose that a function \(f\) defined on some domain \([a,b]\) has the following property: for any \(N\) between \(f(a)\) and \(f(b)\) there exists some \(c\) between \(a\) and \(b\) such that \(f(c) = N.\) Must \(f\) be a continuous function?