Evaluating Limits: x Approach 0 & Beyond

In summary, the limit of ##\sin(x)/x## as x approaches a finite constant can be evaluated using continuity, while the limit as x approaches infinity can be evaluated using the Squeeze Theorem.
  • #1
doktorwho
181
6

Homework Statement


##\frac{e^x-1}{x}##
Evaluate the limit of the expression as x approaches 0.

Homework Equations


3. The Attempt at a Solution [/B]
The question i have is more theoretical. I was able to solve this problem by expanding the expression into the talyor polynomial at ##x=0##. I found that to be the easiest way to get to solution if tou asked to find a limit as x approaches 0. But my question is this:
If x approaches some constant could i do the same thing except evaluate at ##x=a##? And what about if x goes to infinity?
For example ##\frac{sinx}{x}## ##=0## as x goes to inifnity but when you try the expansion it doesnt. We can't then do that, so what would then be the easiest way?
 
Physics news on Phys.org
  • #2
doktorwho said:
what would then be the easiest way
$$\left | {\sin x\over x } \right | \le {\left | {1\over x } \right | }$$
 
  • #3
doktorwho said:
If x approaches some constant could i do the same thing except evaluate at ##x=a##?
Yes. It comes down to a coordinate shift: replace ##x-a## by ##y\ \ \ ## and take ##\ \displaystyle \lim_{y\downarrow 0}##
 
  • #4
doktorwho said:

Homework Statement


##\frac{e^x-1}{x}##
Evaluate the limit of the expression as x approaches 0.

Homework Equations


3. The Attempt at a Solution [/B]
The question i have is more theoretical. I was able to solve this problem by expanding the expression into the talyor polynomial at ##x=0##. I found that to be the easiest way to get to solution if tou asked to find a limit as x approaches 0. But my question is this:
If x approaches some constant could i do the same thing except evaluate at ##x=a##? And what about if x goes to infinity?
For example ##\frac{sinx}{x}## ##=0## as x goes to inifnity but when you try the expansion it doesnt. We can't then do that, so what would then be the easiest way?

Do you mean you want to evaluate ##\lim_{x \to a} \sin(x)/x## for finite ##a > 0##? If so, just use "continuity": ##\sin(x)## is a continuous function, so ##\lim_{x \to a} \sin(x) = \sin(a)## for any finite ##a##, and
$$\lim_{x \to a} \frac{\sin(x)}{x} = \frac{\lim_{x \to a} \sin(x)}{\lim_{x \to a} x}$$
if both limits in the numerator and denominator exist and the denominator limit is not zero.

These are general properties that you should learn because they are used extensively.

As for the limit when ##x \to \infty##, the solution has already been suggested by BVU, but to expand on his/her answer: use the fact that ##-1 \leq \sin(x) \leq 1##, so for ##x>0## we have ##-1/x \leq \sin(x)/x \leq 1/x##. Now use the "Squeeze Theorem"; see, eg.,
https://en.wikipedia.org/wiki/Squeeze_theorem
or
http://www.sosmath.com/calculus/limcon/limcon03/limcon03.html
 
  • Like
Likes doktorwho

FAQ: Evaluating Limits: x Approach 0 & Beyond

What is a limit?

A limit is the value that a function approaches as the input variable gets closer and closer to a specific value, typically denoted as x approaches a certain number or as x approaches infinity.

Why is it important to evaluate limits?

Evaluating limits allows us to determine the behavior of a function at specific points or as the input variable approaches certain values. It helps us understand the continuity, differentiability, and overall behavior of a function.

How do you evaluate limits?

Limits can be evaluated using algebraic techniques such as factoring, simplifying, and rationalizing. They can also be evaluated graphically by looking at the behavior of the function near the point in question. Additionally, we can use numerical methods such as plugging in values closer and closer to the point to estimate the limit.

What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the behavior of the function as the input variable approaches from one side (either the left or the right) of the specified value. A two-sided limit considers the behavior of the function as the input variable approaches from both sides of the specified value.

What does it mean if a limit does not exist?

If a limit does not exist, it means that the function does not have a defined value at the specified point or as the input variable approaches a certain value. This can occur if there are vertical asymptotes, holes, or oscillations in the graph of the function at that point.

Similar threads

Back
Top