Limits are a fundamental concept in calculus, and they can be used to find the value of a function as the input approaches a certain value. One common method for finding limits is to use powers of 10. This method can be used to find the limit of a function as the input approaches any real number, and it is particularly useful for finding the limit of a function as the input approaches infinity.
To use powers of 10 to find the limit of a function, we first need to rewrite the function in terms of powers of 10. This can be done by factoring out the highest power of 10 that divides the function. For example, the function f(x) = x^2 + 3x – 4 can be rewritten as f(x) = (10^0)x^2 + (10^-1)3x – (10^-2)4. Once the function has been rewritten in terms of powers of 10, we can use the following rule to find the limit of the function as the input approaches a certain value:
If lim_(x->a) f(x) = L, then lim_(x->a) (10^n)f(x) = (10^n)L for any integer n.
Factor Out the Powers of 10
In some cases, it’s possible to factor a power of 10 out of the numerator and denominator of the fraction before evaluating the limit. This can simplify the process and make it easier to calculate the limit.
For example, consider the limit:
| lim | x → 0 | (x2 – 16)/(x – 4) |
|---|
We can factor a power of 10 out of the numerator and denominator as follows:
| = lim | x → 0 | (x + 4)(x – 4)/(x – 4) |
|---|---|---|
| = limx → 0(x + 4) | ||
| = 4 |
5. Substituting into the Original Function
Once you’ve evaluated the limit of the simplified fraction, you need to substitute that value back into the original function to get the final answer.
For example, if you have the function:
f(x) = (x2 – 16)/(x – 4)
and you’ve determined that the limit of the function as x approaches 0 is 4, then the value of f(0) is 4.
This can be written as:
| lim | x → 0 | f(x) | = f(0) |
|---|---|---|---|
| = 4 |
How To Find Limit Using Powers Of 10
To find the limit of a rational expression when the denominator approaches a power of 10, rewrite the numerator and the denominator in terms of the same powers of 10 and then simplify.
For example, the limit of (2x – 1)/(x – 5) as x approaches 10 is found by rewriting the expression as
[2(10) – 1]/[(10) – 5] = [19]/[5] = 3.8.
People Also Ask
How to find the limit of a function as x approaches infinity using the powers of 10?
To find the limit of a rational expression as x approaches infinity using the powers of 10, divide both the numerator and the denominator by the highest power of x in the denominator and simplify.
What is the difference between the limit of a function as x approaches a power of 10 and the limit of a function as x approaches infinity?
The limit of a function as x approaches a power of 10 is the value the function approaches as x gets very close to but does not reach the power of 10.
The limit of a function as x approaches infinity is the value the function approaches as x gets larger and larger without bound.
