Solving for Parameter a in a Piecewise Function

In summary, to find all values of a>0 such that the function f(x)=\left\{\begin{array}{cc}\frac{a^x+a^{-x}-2}{x^2},x>0\\3ln(a-x)-2,x\leq0\end{array}\right attains the same value at x=0, we can use L'Hopital's Rule to find the limit of the upper expression, which is \ln^{2}(a)/2. This must equal 3\ln(a)-2 to satisfy the condition. This leads to a quadratic equation in \ln(a), which can be solved to find the values of a that satisfy the condition.
  • #1
azatkgz
186
0

Homework Statement



Find all values of the parameter a>0 such that the function

[tex]f(x)=\left\{\begin{array}{cc}\frac{a^x+a^{-x}-2}{x^2},x>0\\3ln(a-x)-2,x\leq0\end{array}\right[/tex]



The Attempt at a Solution



[tex]\lim_{x\rightarrow 0}\frac{a^x+a^{-x}-2}{x^2}=0[/tex]

[tex]0=3ln(a-0)-2\rightarrow a=e^{2/3}[/tex]
 
Physics news on Phys.org
  • #2
Why should the limit value be 0?? :confused:

All that is required is that AT x=0, both expressions should attain the same value!

Now, consider the function's expression for the non-positives.
Clearly, its limit at x=0 is 3ln(a)-2.
Thus, you are to determine those values of "a" so that the limiting value of the function expression for the positives equals 3ln(a)-2 as well.
 
  • #3
ok
[tex]\lim_{x\rightarrow 0}\frac{a^x+a^{-x}-2}{x^2}=\frac{ln^2a}{2}[/tex]
by L'Hopital Rule

[tex]\frac{ln^2a}{2}=3ln(a)-2[/tex]

[tex]ln^2a-4lna+4=0[/tex]
 
Last edited:
  • #4
The limit of the upper expression should be [tex]\ln^{2}(a)[/tex]

Thus, you have the quadratic in ln(a) to solve:
[tex]\ln^{2}(a)=3\ln(a)-2[/tex]
 

1. What is continuity of a function?

Continuity of a function refers to the property of a mathematical function that allows it to be drawn without any breaks or gaps. In other words, there are no sudden jumps or holes in the graph of a continuous function.

2. How is continuity of a function defined?

The formal definition of continuity of a function is that for any given value of x, the limit of the function as x approaches that value must equal the value of the function at that point. This can also be written as the limit of f(x) as x approaches a is equal to f(a).

3. What are the three types of continuity?

The three types of continuity are: point continuity, where the function is continuous at a specific point; interval continuity, where the function is continuous over a specific interval; and uniform continuity, where the function is continuous over the entire domain.

4. How can we determine if a function is continuous?

A function is considered continuous if it satisfies the following three criteria: 1) the function is defined at the point in question, 2) the limit of the function exists at that point, and 3) the limit is equal to the value of the function at that point.

5. What are some real-life examples of continuity of a function?

Continuity of a function can be observed in many real-life situations, such as the temperature of a room over time, the height of a growing plant, or the speed of a moving car. In these cases, there are no sudden changes or breaks in the data, indicating that the functions describing these phenomena are continuous.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
792
  • Calculus and Beyond Homework Help
Replies
4
Views
518
  • Calculus and Beyond Homework Help
Replies
1
Views
383
  • Calculus and Beyond Homework Help
Replies
8
Views
757
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
939
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
437
  • Calculus and Beyond Homework Help
Replies
7
Views
133
  • Calculus and Beyond Homework Help
Replies
27
Views
607
Back
Top