Chapter 2: Worksheet 3 Jack K. Cohen Colorado School of Mines




Function Composition, Limits, Continuity


Suggested Problems Section 2.4: 2, 6, 8, 22, 26


  1. Recognizing complicated functions as compositions of simpler functions will be important later in this course. For example if h(x) = sin(cos x), we may introduce u = cos x and write h in the simpler form h(x) = sin u. This is equivalent to writing h(x) = f (g(x)), where u = g(x) = cos x and f (x) = sin x. Introduce a useful decomposition for each of the following functions, h(x) using both the u notation and the f, g notation.
    1. sin x3 Note: this notation is common, but perhaps misleading—remember that it means sin(x3), not the function below in part (b).
    2. sin3x Note: this common notation for powers of trigonometric functions is misleading! It often helps to rewrite such expressions in unambiguous notation as (sin x)3.
    3. 3sin x
    4. sin(x3 + 3)
    5. (x4 + x2 +1)7

  2. Extend the above idea to 3-fold decompositions using u, v notation for the intermediate functions. Example: For h(x) = sin(cos(tan x)) introduce u = tan x to get h(x) = sin(cos u) and then v = cos u to finally get h(x) = sin v.
    1. tan(ln(x + 1))
    2. 2sin x2
    3. sin(sin(sin x))

  3. Investigate what happens if we repeatedly compose sin x with itself. That is, we form the sequence sin x, sin(sin x), sin(sin(sin x)),….
    1. Find the limit of this sequence when one puts x = 0 in each term (this is easy).
    2. What about the limit when x = π/2? As a start, compute sin(sin(sinπ/2)). (Or, look at the Plots of the sucessive composed functions).
    Notice that the triple composition of sin with itself is not the same thing as either of the functions discussed in problem 1, parts (a), (b)—all three are different! If you have any doubt of this, Plot each of them.

  4. You have noticed that many limits can be obtained by just evaluating the given function at the given point. State precisely a condition under which this is valid. Which of the following limits can be evaluated in this simple way? Justify your answer.
    1. $\lim_{{x \rightarrow 5}}^{}$ ${\frac{{x + 5}}{{x^4 + x^2 + 1}}}$
    2. $\lim_{{x \rightarrow 5}}^{}$ ${\frac{{x^2 - 25}}{{x - 5}}}$
    3. $\lim_{{x \rightarrow 5}}^{}$$\sqrt{{x^6 - 5}}$

  5. Here are some limits that will play a key role in developing important calculus results. A firmly logical derivation of them is a few weeks away. Meanwhile, employ one of your many rough and ready methods to get the right anwers to engineering accuracy.
    1. $\lim_{{x \rightarrow 0}}^{}$ ${\frac{{\sin x}}{{x }}}$
    2. $\lim_{{x \rightarrow 0}}^{}$ ${\frac{{1 - \cos x}}{{x }}}$
    3. $\lim_{{x \rightarrow 0}}^{}$(1 + x)1/x