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So we do a lot of variable substitution in limits at school. Stuff like $\lim\limits_{x\to5}\ (x+1)^2\ =\ \lim\limits_{y\to6}\ y^2$, where we define the substitution $y = x + 1$.

But I've never been clear on what exactly the theoretical basis for this is. What is the formula that you're actually applying when you do variable substitution? What are the formal conditions under which it is possible?

My conjecture would be the following:

For all continuous $f$, and all real $a$:
$\lim\limits_{x\to a}\ (f\circ g)(x)\ =\lim\limits_{x\to g(a)}\ f(x)$, where $g$ is a continuous function

So to take my first example, $f$ would be $x^2$, $g$ would be $x + 1$, and $a$ would be $5$.

Am I in the right area? If this is correct, can it be proven using $\epsilon$-$\delta$? I had a half-hearted shot at it the other night and didn't get anywhere.

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6 Answers 6

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The complete story is as follows:

If the functions $g:\ A\to B$ and $f:\ B\to C$ have limits $$\lim_{x\to\xi}g(x)=:\eta\ ,\qquad \lim_{y\to\eta}f(y)=:\zeta\ ,$$ and if $f$ is continuous at $\eta$ in case $\eta$ occurs as value of $g$, then $$\lim_{x\to\xi}f\bigl(g(x)\bigr)=\lim_{y\to\eta} f(y)\ .$$

This holds also if any one of $\xi$, $\eta$, $\zeta$ is $\ =\infty$.

The extra condition "and if $f$ is continuous $\ldots$" is usually fulfilled, but one cannot do without it: Consider the example $g(x):\equiv 1$ and $f(y):=2$ $\ (y=1)$, $\ f(y):=3$ $\ (y\ne1)$. Then $\lim_{x\to1}f\bigl(g(x)\bigr)=2$, but $\lim_{x\to1}g(x)=1$, $\ \lim_{y\to1}f(y)=3$.

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  • $\begingroup$ So essentially, in this version, by using the limit of f rather than its exact value at a particular point, we can include cases at infinity in the theorem? $\endgroup$
    – Jack M
    Jul 7, 2012 at 19:00
  • $\begingroup$ @Jack M: Yes. By the way, I have changed the rôles of $f$ and $g$, as suggested by Arturo Magidin. $\endgroup$ Jul 7, 2012 at 19:40
  • $\begingroup$ I found this answer those most clear and complete. Arturo did include a proof, but this answer covers limits at infinity (to be fair though, my wording could be interpreted as only covering real limits). Also I was able to work out why a variable substitution I tried the other day failed - $f$ wasn't continuous at the limit of $g$. $\endgroup$
    – Jack M
    Jul 7, 2012 at 21:58
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    $\begingroup$ 1) There is no requirement the value of $g(\xi)$ right? It can be any real number? 2) If $g(x)$ does not equal $\eta$ except possibly at $\xi$, then continuity of $f$ at $\eta$ is not required, correct? Thanks. $\endgroup$
    – TSJ
    Mar 22, 2015 at 3:16
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    $\begingroup$ One more comment. The second condition states we need to know the original limit, $\lim_{y\to\eta}f(y)$, actually exists. So there are cases where after the change of variable we find $\lim_{x\to\xi}f\bigl(g(x)\bigr)$ exists, but the original $\lim_{y\to\eta}f(y)$ does not, in fact, exist? It would really useful in practice if we can say, under some condition, $\lim_{x\to\xi}f\bigl(g(x)\bigr)$ exists iff $\lim_{y\to\eta}f(y)$ exists. $\endgroup$
    – TSJ
    Mar 22, 2015 at 3:34
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Yes, that's essentially the idea.

But this follows from the definition of continuity: we just need to show that if $f$ is continuous at $g(a)$ and $g$ is continuous at $a$, then $f\circ g$ is continuous at $a$. Because then both sides evaluate to $f(g(a))$, by definition of continuity (which requires that the limit exist and be equal to evaluating the function at the point.

To prove that if $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$, note first that $f\circ g$ is defined at $a$. Now, let $\epsilon\gt0$. then we know that there exists $\delta_1\gt 0$ such that if $|y-g(a)|\lt\delta_1$, then $|f(y)-f(g(a))|\lt\epsilon$; this holds, because $f$ is continuous at $g(a)$.

Now, since $g$ is continuous at $a$, and $\delta_1\gt 0$, this means that there exists $\delta\gt 0$ such that if $|x-a|\lt\delta$, then $|g(x)-g(a)|\lt\delta_1$.

Thus, if $|x-a|\lt\delta$, then $|g(x)-g(a)|\lt\delta_1$, and therefore $|f(g(x))-f(g(a))|=|f\circ g(x) - f\circ g(a)|\lt\epsilon$.

Thus, for all $\epsilon\gt 0$ there exists $\delta\gt 0$ such that if $|x-a|\lt\delta$, then $|f\circ g(x) - f\circ g(a)|\lt\epsilon$. Therefore, $f\circ g$ is continuous at $a$.

Therefore, we have that $\lim\limits_{y\to g(a)} f(y) = f(g(a))$, since $f$ is continuous at $g(a)$; and $\lim\limits_{x\to a}f\circ g(x) = f\circ g(a) = f(g(a))$, since $f\circ g$ is continuous at $a$.

You don't quite need $g$ to be continuous: if $\lim\limits_{x\to a}g(x)=L$ and $f$ is continuous at $L$, then we have $$\lim_{x\to a}f\circ g(x) = \lim_{y\to L}f(y) = f(L).$$ To verify this, let $\epsilon\gt 0$. Then there exists $\delta_1\gt 0$ such that for all $x$, $|y-L|\lt\delta_1$ implies $|f(y)-f(L)|\lt\epsilon$. Since $\lim\limits_{x\to a}g(x)=L$, there exists $\delta\gt 0$ such that if $0\lt |x-a|\lt \delta$ then $|g(x)-L|\lt\delta_1$. So, suppose that $0\lt |x-a|\lt\delta$. Then $|g(x)-L|\lt\delta_1$, and therefore $|f(g(x))-f(L)|\lt\epsilon$. Therefore, for every $\epsilon\gt 0$ there exists $\delta\gt 0$ such that if $0\lt |x-a|\lt\delta$, then $|f(x)-f(L)|\lt\epsilon$. This proves that if $\lim\limits_{x\to a}g(x) = L$ and $f$ is continuous at $L$, then $$\lim\limits_{x\to a}f\circ g(x) = \lim\limits_{y\to L}f(y) = f(L).$$ In particular, if $g$ is continuous at $a$, then we replace $L$ with $g(a)$.

We cannot omit the continuity of $f$ at $L$, though: take $g(x) = 0$ for all $x$, and let $$f(x) = \left\{\begin{array}{ll} 1 &\text{if }x\neq 0\\ 0 &\text{if }x=0. \end{array}\right.$$ Then $\lim\limits_{x\to a}f(g(x)) = 0$, because $f(g(x))=f(0)=0$. But $$\lim\limits_{y\to 0}f(y) = 1,$$ because we never take the value $y=0$ in evaluating the limit.

One can replace continuity of $f$ with other conditions; for example, we may ask that $g$ have a limit $L$ at $a$, and moreover, that for every $\delta\gt0$ there exist an $\eta\gt 0$ such that $g$ takes all values on $(L-\eta,L+\eta)$, except perhaps $L$ itself, on $(a-\delta,a+\delta)-\{a\}$.

Added. The situation with limits as $x\to\infty$ is essentially the same if $\lim\limits_{x\to\infty}g(x)$ exists and is real. It is more complicated when the limit of $g$ is $\pm\infty$. See this answer for some discussion on that.

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    $\begingroup$ @BabakSorouh: $g$ not being $0$ near $x_0$ is completely and utterly irrelevant. $0$ is a red herring. We are not taking quotients, we are composing, so values of $0$ are irrelevant. Just add $25$ to the values I gave: take $g$ to be constant $25$, take $f$ to be $25$ at $25$ and $26$ elseewhere. Then $g$ has limit $a=25$ at $x_0=25$, and $f$ has limit $A=26$ at $a=25$. But $\lim\limits_{x\to 25}f\circ g(x)=25$ and $\lim\limits_{y\to 25}f(y) = 26$. $\endgroup$ Jul 7, 2012 at 18:53
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    $\begingroup$ @William: The paragraph at the end, before "Added:" explicitly says you can replace continuity with other conditions, and the situation you specify is included in the sample condition I gave there 9 years ago. $\endgroup$ Nov 11, 2021 at 14:26
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    $\begingroup$ @William:The idea there is just that when you are trying to find the limit of $f(g(x))$, things may "work out" because $g$ "skips over" any bad points that $f$ may have near $L=\lim g(x)$. And that might mess you up when you try to do $\lim_{x\to L}f(x)$. The condition I give ensures that the composition "hides nothing" that is happening near $L$ for $f$. Continuity of $f$ at $L$ similarly ensures that there is nothing weird for $f$ near $L$ that may be hidden by the composition. $\endgroup$ Jan 16 at 23:05
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    $\begingroup$ @William: What the condition implies is that $g$ is always surjective near near $L$; so that when you look at the limit of the composition, it is always forcing you to consider all values that $f$ takes for inputs near $L$. That is precisely what you need to do when you are trying to find the limit $\lim_{x\to L}f(x)$, but not necessarily when you are trying to find the limit $\lim_{x\to a}f(g(x))$. $\endgroup$ Jan 16 at 23:09
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    $\begingroup$ @William: The main difference between $\lim_{x\to a}f(g(x))$ and $\lim_{x\to L}f(x)$, when $\lim_{x\to a}g(x)=L$, is that in the first you only need to consider the values that $f(x)$ takes near $L$ in the range of $g(x)$, whereas in the latter you need to consider the values that $f(x)$ takes everywhere near $L$. Any condition that ensures wither that "all values near $L$" are also in the range of $g(x)$, or that just looking at the ones in the range of $g$ will not mess you up, will give you equality of limits. $\endgroup$ Jan 16 at 23:16
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While this is a fairly old question, I stumbled upon it and wanted to give a different take for the case where the continuity of $f$ cannot be assumed. To make things work, then, we need to strengthen the hypotheses on $g$.

Theorem. Suppose $g$ is continuous and injective on an open interval $I$ containing $a$ and $f$ is some function. Then $\displaystyle \lim_{x \to a}{(f \circ g)(x)}$ exists if and only if $\displaystyle \lim_{t \to g(a)}{f(t)}$ exists, and they are equal when they do exist.

Proof. Suppose $\displaystyle \lim_{x \to a}{f(g(x))} = L$. We claim that $\displaystyle \lim_{t \to g(a)}{f(t)} = L$. To that effect, let $\epsilon>0$ be arbitrary. Our hypothesis means that there is a $\delta > 0$ such that $$0 < |x-a| < \delta \implies |f(g(x)) - L| < \epsilon.$$ Because $g$ is one-to-one and continuous on an open interval $I$, $g^{-1}$ is also one-to-one and continuous on the open interval $g[I]$. Thus, there exists a $\gamma>0$ such that $$|t - g(a)| < \gamma \implies |g^{-1}(t) - a| < \delta.$$ Because $g^{-1}(t) = a$ only occurs when $t=g(a)$, we similarly have $$0 < |t - g(a)| < \gamma \implies 0 < |g^{-1}(t) - a| < \delta.$$ Thus, for all $t$, $$0 < |t - g(a)| < \gamma \implies 0 < |g^{-1}(t) - a| < \delta \implies |f(g(g^{-1}(t))) - L| < \epsilon$$ or $$0 < |t - g(a)| < \gamma \implies |f(t) - L| < \epsilon.$$ Since $\epsilon>0$ was arbitrary, it follows that $\displaystyle \lim_{t \to g(a)}{f(t)} = L$.

The opposite direction follows from the above direction: if we let $h = f \circ g$ and $b=g(a)$, then $\displaystyle \lim_{t \to g(a)}{f(t)}$ may be written as $\displaystyle \lim_{t \to b}{h(g^{-1}(t))}$, and $\displaystyle \lim_{x \to a}{f(g(x))}$ may be written as $\displaystyle \lim_{x \to g^{-1}(b)}{h(x)}$. Since $g^{-1}$ satisfies the same hypotheses as $g$ and there were no requirements on $f$, the above argument goes through in this new arrangement. Q.E.D.

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Here is my point of view. Let $f:\text{D}_f\to\mathbb{R}$ and $g:\text{D}_g\to \text{R}_g\subseteq\text{D}_f$ be functions so that $f\circ g:\text{D}_g\to\mathbb{R}$ is well defined. Well, I assume that you are aware of the following theorem.

Theorem 1. If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$ then $f\circ g$ is continuous at $a$. In terms of limit notation, if $\lim_{x\to a}g(x)=g(a)$ and $\lim_{x\to g(a)}f(x)=f(g(a))$ then we have $\lim_{x\to a}(f\circ g)(x)=(f\circ g)(a)$.

A slight generalization of this theorem is the following

Theorem 2. Suppose that $\lim_{x\to a}g(x)=b$ and $\lim_{x\to b}f(x)=c$. Furthemore, assume that $g$ satisfies the following condition $$\exists r>0,\,\,\,0<|x-a|<r\implies 0<|g(x)-b|.$$ Then we have $\lim_{x\to a}(f\circ g)(x)=c$.

In your example, we have $f:x\mapsto x^2$ and $g:x\mapsto x+1$, $a=5$, $b=6$. We clearly have $\lim_{x\to5}x+1=6$ and $\lim_{x\to 6}x^2=36$. More importantly, choose $r=1$ so whenever $0<|x-5|<1$ we have that $|(x+1)-6|>0$. These two imply that $\lim_{x\to 6}(x+1)^2=36$. In practice, one usually write this as

$$\lim_{x\to 5}(x+1)^2=\lim_{y\to 6}y^2=36.$$

What happens in the mind of calculus students is that they detect $g:x\mapsto x+1$ instantly, compute $\lim_{x\to 5}x+1=6$ in their (unconscious) mind and get the first equality which is then easily computed to get the second one. One tricky point about this is that as you do the writing from left to right, you may think that the implications are also in this order while it is exactly in the converse direction, right to left! Also, $y$ here is just a dummy variable and can be replaced with other variables such as $t$ or the $x$ itself.

We can also express another variation of Theorem 2 as below.

Theorem 3. Suppose that $\lim_{x\to a}g(x)=b$ and $\lim_{x\to b}f(x)=c$. Furthemore, assume that $f$ is continuous at $b$. Then we have $\lim_{x\to a}(f\circ g)(x)=c$.

Remark 1. Sometimes, instead of writing $\lim_{x\to a}(f\circ g)(x)=c$, we use the more intuitive symbolism of $\lim_{g(x)\to b}f(g(x))=c$.

Remark 2. These theorems can naturally be generalized for metric spaces.

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Let $L_1 = \lim\limits_{x \to g(a)} f(x)$ and $L_2 = \lim\limits_{x \to a} f(g(x))$. $L_1$ is unique in $\mathbb{R}$ so that for all $\varepsilon > 0$, there exists $\delta_1(\varepsilon) > 0$ such that if $|x-g(a)| < \delta_1(\varepsilon)$ then $|f(x) - L_1| < \varepsilon$. Similarly $L_2$ is unique so that there exists $\delta_2(\varepsilon) > 0$ such that if $|x-a| < \delta_2(\varepsilon)$ then $|f(g(x)) - L_2| < \varepsilon$.

Now we will show that $L_1$ also satisfies the property that $L_2$ satisfies uniquely. Take $\varepsilon > 0$. When $|x-g(a)| < \delta_1(\varepsilon/2)$,

\begin{align*} |f(g(x)) - L_1| &= |f(g(x)) + f(x) - f(x) - L_1| \\ &\le |f(g(x)) - f(x)| + |f(x) - L_1| \\ &< \varepsilon/2 + |f(g(x) - f(x)|. \end{align*}

Since $f$ is continuous, there exists $\delta(\varepsilon/2) > 0$ such that when $|x-g(a)| < \delta(\varepsilon/2)$, $|f(g(x)) - f(x)| < \varepsilon / 2$. Thus if $|x - g(a)| < \min\{\delta_1(\varepsilon/2), \delta(\varepsilon/2)\}$, then we get \begin{align*} |f(g(x)) - L_1| < \varepsilon/2 + \varepsilon/2 = \varepsilon. \end{align*} Thus $L_1$ is also the limit $\lim\limits_{x\to a} f(g(x))$ and by uniqueness we conclude $L_1 = L_2$, i.e. $\lim\limits_{x\to a} f(g(x)) = \lim\limits_{x\to g(a)} f(x)$.

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@ChristianBlatter 's answer is not completely right. $f$ does not need to be continuous.

Now we show the strengthened proposition:

Let $f:A \to C$, $g: B \to C$, and $\varphi: B \to A$ be three functions. If $\lim_{x \to a}f(x) = L$, $\lim_{x \to b}\varphi(x) = a$, and $g(x) = f(\varphi(x))$, then $g$ has the limit $L$ as $x$ approaches $b$.

Proof. Since that $g(x) = f(\varphi(x))$, $|g(x) - L| < \varepsilon \equiv |f(\varphi(x)) - L| < \varepsilon$. For every $\varepsilon>0$, we can find $\delta>0$ such that $|\varphi(x)-a|<\delta \Longrightarrow |f(\varphi(x)) - L| < \varepsilon$, as $f$ has a limit. Furthermore, for this $\delta$, we can always find $\delta'$ such that $|x-b|<\delta' \Longrightarrow |\varphi(x)-a|<\delta$ because $\varphi$ also has a limit.

Then for every $\varepsilon >0$, we can find $\delta'>0$ such that $$|x-b|<\delta' \Longrightarrow |g(x) -L|<\varepsilon$$, as desired.

Note that this is essentially saying that $\lim_{x \to b}f(\varphi(x))=L$, since $g=f\circ \varphi$.

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