Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The question is:

Suppose $\lim_{x\rightarrow 1}\frac{f(x)-7}{x-1}=4$, find $\lim_{x\rightarrow 1}f(x)$.

The obvious answer is $7$, by going: $$\begin{align*} \lim_{x\rightarrow 1}\frac{f(x)-7}{x-1}&=4\\ \Rightarrow\lim_{x\rightarrow 1}[f(x)-7]&=4\times{\lim_{x\rightarrow 1}[x-1]}\\ \Rightarrow\lim_{x\rightarrow 1}[f(x)-7]&=0\\ \Rightarrow\lim_{x\rightarrow 1}f(x)&=7 \end{align*}$$ But this feels wrong. Specifically, the step of taking "${\lim_{x\rightarrow 1}[x-1]}$" over to the right hand side seems illegal. I justify this to myself by saying we can treat it like a number not equal to zero because it is a limit. But I don't believe my justification.

share|cite|improve this question
up vote 7 down vote accepted

Anwser to yout "seems illegal" doubt: as $$\lim_{x\to 1}\frac{f(x)-7}{x-1}\text{ exists}$$ and $$\lim_{x\to 1}(x-1)\text{ exists},$$ $$\lim_{x\to 1}(f(x)-7) = \lim_{x\to 1}\frac{f(x)-7}{x-1}\times\lim_{x\to 1}(x-1) = \cdots\text{ exists.}$$

share|cite|improve this answer
    
Oh! Of course! Silly me for not writing all my working! Thanks. – user303353 Jan 7 at 11:26

Hint: Its a matter of rewrite $f(x) = (x-1)\cdot \dfrac{f(x)-7}{x-1} + 7\Rightarrow \displaystyle \lim_{x\to 1} f(x) = \displaystyle \lim_{x\to1}(x-1)\cdot \displaystyle \lim_{x\to 1}\dfrac{f(x)-7}{x-1} + \displaystyle \lim_{x \to 1} 7 = ?$

share|cite|improve this answer
    
So $7$ is correct then? – user303353 Jan 7 at 11:17
1  
Yes. But I am confused because I give the answer as $7$ in my question, so am wondering why you didn't just say "Yes, the answer is $7$. Another way to see this is by..." or "Yes, the answer is $7$. Your way is incorrect because... A better way to get the answer is by...". So I am wondering if I am missing something? – user303353 Jan 7 at 11:23

The subtlety of this question is where the 4 comes from, since $f(1)$ must be 7 for the limit to exist. This quotient is the definition of derivitive so what it's saying is that $f'(1) = 4.$

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.