# Untitled Page 19

- Page ID
- 148358

## Homework Problems

**1**.
(a) Prove, using the Weierstrass definition of the limit, that if \lim_{x→ a} f(x) = F and \lim_{x→ a} g(x) = G both exist,
them \lim_{x→ a} [f(x)+g(x)] = F+G, i.e., that the limit of a sum is the sum of the limits. (b) Prove the same thing using the
definition of the limit in terms of infinitesimals.
(solution in the pdf version of the book)

**2**.
Sketch the graph of the function *e*^{-1/x}, and evaluate the following four limits:

(solution in the pdf version of the book)

**3**.
Verify the following limits.

[Granville, 1911] (solution in the pdf version of the book)

**4**.
Evaluate

exactly, and check your result by numerical approximation. (solution in the pdf version of the book)

**5**.
Amy is asked to evaluate

She applies l'H\^{o}pital's rule, differentiating top and bottom to find
1/*e*^{x}, which equals 1 when she plugs in *x*=0. What is wrong with her reasoning?
(solution in the pdf version of the book)

**6**.
Evaluate

exactly, and check your result by numerical approximation. (solution in the pdf version of the book)

**7**.
Evaluate

exactly, and check your result by numerical approximation. (solution in the pdf version of the book)

**8**.
Prove a form of l'H\^{o}pital's rule stating that

is equal to the limit of *f*'/*g*' at infinity. Hint: change to some
new variable *u* such that *x*→∞ corresponds to *u*→0.
(solution in the pdf version of the book)

**9**.
Prove that the linear function *y*=*ax*+*b*, where *a* and *b* are real, is continuous, first using the definition of
continuity in terms of infinitesimals, and then using the definition in terms of
the Weierstrass limit.
(solution in the pdf version of the book)

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