Limit involving exponential function question

In summary, the conversation discusses finding the limit as x tends to zero of a given expression. The suggested method of Taylor expanding the exponential and cosine functions makes finding the limit easier.
  • #1
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Homework Statement


Find the limit as x tends to zero of: (e^-x - cos x)/2x


Homework Equations


lim_x->0 e^-x = 1
lim_x->0 cos x = 1
lim_x->0 sin x / x= 1


The Attempt at a Solution


Hi everyone,
Here's what I've done so far:

(e^-x - cosx)/2x = [(e^-x)^2 - (cosx)^2] / 2x(e^-x - cosx) ... multiplying by conjugate

= [e^-2x - 1 + (sinx)^2 ] / 2x(e^-x - cosx)
... And so I want to try and isolate the sinx to put it over the x, which will then go to 1. But I don't know how to do this. Or am I going about it the wrong way entirely?

Thanks for any help
 
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  • #2
I don't think that is going to work. You could taylor expand both the exponential and the cosine. This allows you to find the limit pretty easily.
 

1. What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = ab^x, where a and b are constants and b is the base. It is often used to model growth or decay in various real-world situations.

2. How do you solve a limit involving an exponential function?

To solve a limit involving an exponential function, you can use the properties of limits and logarithms. First, rewrite the function using logarithms. Then, use the limit laws to simplify the expression. Finally, evaluate the limit using algebraic techniques.

3. What are the common mistakes to avoid when solving a limit involving an exponential function?

Some common mistakes to avoid when solving a limit involving an exponential function include forgetting to use logarithms, not simplifying the expression before evaluating the limit, and using incorrect limit laws.

4. Can a limit involving an exponential function have multiple solutions?

Yes, a limit involving an exponential function can have multiple solutions. This can happen if the limit approaches different values from the left and right sides, or if there are discontinuities in the function.

5. How can understanding limits involving exponential functions be applied in real life?

Understanding limits involving exponential functions can be applied in real life to model and predict growth or decay in various fields such as finance, biology, and physics. It can also be used to optimize processes and make informed decisions based on data.

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