# Need some help with limits and continuity

1. Oct 13, 2011

### dustinm

I have 2 questions in regards to continuity and limits.

Question 1:
f(x)= e$^{-x^{2}}$ if x ≠ 0.
f(x)= c if x=0.

For which value of c is f(x) continuous at x=0?

I was thinking the answer would be 1 but I feel that's incorrect.

Question 2:
Compute lim x→∞f(x).

I'm not familiar with how to solve limits to infinity when the variable is in the exponent.
Any and all help is appreciated! thank you guys.

2. Oct 13, 2011

### gb7nash

Ok, what does it mean to be continuous at x = 0? Look at your definitions.

What does the exponent of e approach as x goes to ∞?

3. Oct 13, 2011

### dustinm

It means that f(x)=f(c) as x→c. So basically that the function can be drawn without having to lift the pen to complete the graph.

So, e$^{-∞^{2}}$ would be approaching -∞?

4. Oct 13, 2011

### gb7nash

Almost. f(x)→f(c) as x→c. In other words:

$$\lim_{x \to c}f(x) = f(c)$$

So we need to look at the left side and the right side of the equation. Replacing c with 0, what is $\lim_{x \to c}f(x)$? What is f(c)? Are they equal? If so...

You're not thinking this all the way through. The exponent is approaching -∞ like you said. What's e raised to a large negative number?

5. Oct 13, 2011

### dustinm

f(c)=0 and if you were to replace that into the equation for f(x) you would get f(x)=e$^{-0^{2}}$ which would end up equaling 1, right?

Sorry about this, all of this is really new to me so it's tough to grasp at first.

Ahh so it would be 0 because e$^{-∞}$ is extremely small.

6. Oct 13, 2011

### gb7nash

Correct (I fixed a typo of yours). That's the right side of the equation. Now you need to look at the left side, which is:

$$\lim_{x \to 0}e^{-x^2}$$

Is this also equal to 1?

Correct.

7. Oct 13, 2011

### dustinm

Yes that would be 1.
So the final answer for making the graph continuous at 0 needs to be 1?

8. Oct 13, 2011

### gb7nash

Correct.

9. Oct 13, 2011

### dustinm

Thank you very much for the help with these questions!!
Walking me through it helped out a bunch!