Bonus problems.

1.

(5pts) Suppose that $\lim_{x \rightarrow a} f(x)$ exists, and $\lim_{x \rightarrow a} g(x)$ does not exist. What can one say about $\lim_{\mathop x\rightarrow a} (f(x)+g(x))$?

2.

(5pts) Is the function

$$f(x)=\left\{ \begin{tabular}{ll} $x\sin \frac{1}{x}$,& $x\ne 0$,\\ $0$, & $x=0$. \end{tabular}\right.$$

continuos on $(-\infty ,\infty )$? Differentiable on $(-\infty ,\infty )$?

3.

(5pts) The same question about

$$f(x)=\left\{ \begin{tabular}{ll} $x^2 \sin \frac{1}{x}$,& $x\ne 0$,\\ $0$, & $x=0$. \end{tabular}\right.$$

4.

(Done) Find all tangent lines to the curve y=x²+x+1 that go through the point (1,1).

5.

(Done) Show that there is no tangent line to the curve y=x²+x+1 that goes through the point (4,1).

6.

(Done) Suppose that we know that the functions $\sin x$ and $\cos x$ are differentiable, but we don't know their derivatives. Using some trigonometric identities, for example,

$$\cos(\alpha -\beta )=\cos \alpha cos \beta + \sin\alpha \sin \beta,$$

$$\cos^2 \alpha +\sin^2 \alpha =1,$$

find the formulas for the derivatives of $\sin x$ and $\cos x$.

7.

(Done) Find the sum

$$1+x+x^2+\ldots\quad +x^n.$$

8.

(Done) Find the sum

$$1+2x+3x^2+\ldots\quad +nx^{n-1}.$$

9.

(5pts) Suppose that $\lim_{x \rightarrow a} f(x)$ and $\lim_{x \rightarrow a} g(x)$ do not exist. What can you say about $\lim_{\mathop x\rightarrow a} (f(x)+g(x))$? Give some examples.

10.

(6pts) Suppose that for some function f

$$\lim_{\mathop x\rightarrow 0} \left( f(x)+\frac{1}{f(x)}\right) =2.$$

Show that $\lim_{x\rightarrow 0} f(x)=1$.

11.

(5pts) Find the limit

$$\lim_{\mathop x\rightarrow 0} \frac{\sqrt[3]{2+\sin x}-\sqrt[3]{2+\tan x}}{x^3}.$$

12.

(Done) If f is differentiable, find

$$\lim_{\mathop x\rightarrow a} \frac{f(x)-f(a)}{x^3-a^3}.$$

13.

(5pts) Let a function f be differentiable at a. Find

$$\lim_{h\rightarrow 0} \frac{f(a+h)-f(a-h)}{h}.$$

14.

(5pts) Find the two points on the curve y=x⁴-2x²-x that have a common tangent line.