Limits, Continuity and Differentiability — Practice

Question 1

Evaluate $\displaystyle\lim_{x	o0}\frac{	an x-\sin x}{x^{3}}$.

Accepted Answer

$	an x=x+	frac{x^{3}}{3}+\cdots$ and $\sin x=x-	frac{x^{3}}{6}+\cdots$

Difference: $	frac{x^{3}}{3}+	frac{x^{3}}{6}=	frac{x^{3}}{2}$.

Limit $=	frac12$.

Question 2

Evaluate $\displaystyle\lim_{x	o0}\left(\frac{\sin x}{x}ight)^{1/x^{2}}$.

Accepted Answer

$f-1=\dfrac{\sin x-x}{x}\approx-\dfrac{x^{2}}{6}$.

$g(f-1)=\dfrac{1}{x^{2}}\cdot\left(-\dfrac{x^{2}}{6}ight)	o-\dfrac16$.

Limit $=e^{-1/6}$.

Question 3

Evaluate $\displaystyle\lim_{x	o0}x^{2}\left[\frac{1}{x}ight]$, where $[\,\cdot\,]$ is the greatest integer function.

Accepted Answer

$\dfrac1x-1<\left[\dfrac1xight]\le\dfrac1x$.

For $x>0$, multiplying by $x^{2}$ gives $x-x^{2}<x^{2}\left[	frac1xight]\le x$, and both bounds tend to $0$.

Repeating for $x<0$ reverses the inequalities but gives the same limit, so the answer is $0$.

Question 4

Find $a$ and $b$ so that $f(x)=\dfrac{\sqrt{1+ax}-\sqrt{1-ax}}{x}$ for $x<0$, $f(0)=b$, and $f(x)=\dfrac{2x+1}{x-2}$ for $x\ge0$ is continuous at the origin.

Accepted Answer

Left limit: multiply by the conjugate to get $\dfrac{2ax}{x\left(\sqrt{1+ax}+\sqrt{1-ax}ight)}	o a$.

Right value at $0$: $\dfrac{1}{-2}=-	frac12$.

Continuity needs $a=b=-	frac12$.

Question 5

Classify the discontinuity of $f(x)=\dfrac{x^{2}-9}{x-3}$ at $x=3$, and of $g(x)=\dfrac{|x-3|}{x-3}$ at the same point.

Accepted Answer

$f(x)=x+3$ for $x
eq3$, so the limit is $6$ while $f(3)$ is undefined: a removable discontinuity.

$g$ has left limit $-1$ and right limit $+1$: a jump discontinuity, which no redefinition at a single point can repair.

Question 6

Where is $f(x)=\left|x^{2}-4\right|$ not differentiable?

Accepted Answer

Zeros of $x^{2}-4$ are at $x=\pm2$.

The derivative of $x^{2}-4$ there is $\pm4
eq0$, so the graph crosses transversally at both.

Hence $f$ fails to be differentiable at exactly $x=2$ and $x=-2$.

Question 7

Show that $f(x)=x\sin\dfrac1x$ for $x
eq0$ with $f(0)=0$ is continuous but not differentiable at the origin.

Accepted Answer

$\left|x\sin	frac1xight|\le|x|	o0=f(0)$, so $f$ is continuous.

$\dfrac{f(h)-f(0)}{h}=\sin\dfrac1h$, which oscillates between $-1$ and $1$ as $h	o0$ and has no limit.

So $f'(0)$ does not exist. One extra factor of $x$ would repair this.

Question 8

Discuss the differentiability of $f(x)=\max\left(x,x^{2}\right)$ on $\mathbb{R}$.

Accepted Answer

They meet at $x=0$ and $x=1$.

At $x=0$ the slopes are $1$ and $0$, so the crossing is transversal: a corner.

At $x=1$ the slopes are $1$ and $2$, again different: a second corner.

So $f$ is differentiable except at $x=0$ and $x=1$.

Question 9

A differentiable function satisfies $f(xy)=f(x)+f(y)$ for all positive $x,y$, with $f'(1)=3$. Find $f$.

Accepted Answer

Setting $x=y=1$ gives $f(1)=0$.

From the definition, $f'(x)=\lim_{h	o0}\dfrac{f\left(x\left(1+	frac hxight)ight)-f(x)}{h}=\lim\dfrac{f\left(1+	frac hxight)}{h}=\dfrac{f'(1)}{x}$.

So $f'(x)=\dfrac3x$ and $f(x)=3\ln x$, which satisfies $f(1)=0$.

Question 10

Prove that $\left|	an^{-1}a-	an^{-1}bight|\le|a-b|$ for all reals.

Accepted Answer

By the mean value theorem there is a $c$ between $a$ and $b$ with $	an^{-1}a-	an^{-1}b=\dfrac{a-b}{1+c^{2}}$.

Since $1+c^{2}\ge1$, the factor has modulus at most $1$.

Taking moduli gives the result.

Question 11

Show that $2x^{3}-9x^{2}+12x+k=0$ has at most one root in $(0,1)$.

Accepted Answer

If there were two roots, Rolle's theorem would give a $c$ in $(0,1)$ with $6c^{2}-18c+12=0$, that is $c^{2}-3c+2=0$.

The roots are $c=1$ and $c=2$, neither of which lies inside $(0,1)$.

So two roots are impossible.

Question 12

If $f$ is continuous on $[0,1]$ with $f(0)=f(1)$, prove that $f(c)=f\left(c+	frac12ight)$ for some $c$ in $\left[0,	frac12ight]$.

Accepted Answer

Let $g(x)=f\left(x+	frac12ight)-f(x)$ on $\left[0,	frac12ight]$, which is continuous.

$g(0)=f\left(	frac12ight)-f(0)$ and $g\left(	frac12ight)=f(1)-f\left(	frac12ight)=f(0)-f\left(	frac12ight)$.

These are negatives of each other, so either both are zero or they have opposite signs. In each case $g$ has a zero, which is the required $c$.

Practice — Limits, Continuity and Differentiability

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