How to Evaluate Limits to Infinity Using Rules and Examples
Sometimes we can't find the actual value of a function directly, but we can see what it should be as we get closer and closer to that particular value. A limit is a value that a function (or a sequence) "approaches" as the input (or index) "approaches" some value.
Limits
Let’s understand this with an example.
Consider the function x2 - 1/ x - 1
We know that the given function is not defined when the value of x is 1, because division by zero is not a valid mathematical operation. So, we can find value as x approaches 1.
Let’s try approaching when x tends to 1.
From the above example, we can see that when x gets close to 1, then the value of x2 - 1/ x - 1 gets close to 2.
When x = 1, we don’t know the value (as it is indeterminate form). We can see that value gets close to 2. We can give answers as 2 but it is not the actual value. Hence the concept of the word “limit” came into existence.
The value of x2 - 1/ x - 1 as x approaches 1 is 2.
Infinity:
Something that is boundless or endless or else something that is larger than any real number is known as infinity. It is denoted by a symbol ∞.
Let’s understand with an example.
Find the value of one divided by infinity (1/∞).
Infinity is not a defined value. So, 1/∞ is similar to 1/smart.
We could say that 1/∞ = 0, but how it is possible because if we try to divide 1 into infinite pieces they can end up to 0 each. You may think about what happened to 1.
In fact, the value of 1/∞ is known to be undefined.
So, instead of trying to find it for infinity. Let’s try it for a larger value.
From the above table, we can see that as x gets larger, 1/x tends to 0.
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We can conclude two fact:
We can’t say what happens to x when it tends to infinity.
We can see that 1/x is getting close to 0.
Hence the limit of 1/x as x approaches infinity is 0. We can write it as
lim (1/x) = 0 when x approaching ∞.
In a mathematical way, we are not talking about when x = ∞, but we know the value as x gets bigger the value gets closer and closer to 0.
So, infinity can’t be used directly but we can use the limit.
Limits to Infinity:
How to find the limit of a function as x approaches infinity?
Let function be y = 2x.
So, from the above table, we can say that as x approaches infinity, then 2x also approaches infinity.
Don’t consider “=” sign as the exact value in the limit. We can’t actually get to infinity, but in limit language the limit is infinity.
Infinity and Degree
Functions like 1/x approaches to infinity. This is also valid for 1/x2 and so on.
A function such as x will approach infinity, same we can apply for 2x or x/9, and so on. Likewise functions with x2 or x3 etc will also approach infinity.
We should be careful with negative functions like -x will approach -infinity. So we have to look at the sign of x and then decide the function value.
Let’s understand negative function value with an example:
Consider 2x2 - 4x
Sol: We know that 2x2 will tend towards +infinity and -5x will tend towards -infinity. But x2 value will be larger as compared to x. So 2x2 - 4x will tend to +infinity.
When we look for the degree of the function, check the highest exponent in the function.
The degree of function is divided into two parts:
The degree is greater than 0, the limit is infinity.
The degree is less than 0, the limit is 0.
Rational Function
A rational function is one that is the ratio of two polynomial functions.
Let f(x) = P(x)/Q(x)
P(x) = x3 + 2x - 1 and Q(x) = 3x2
Now compare the degree of P(x) to the degree of Q(x).
If the degree of P is less than the degree of Q the limit value is 0.
If the degree of P and Q are the same divide the coefficient of terms with the largest exponent.
If the degree of P is greater than the degree of Q then two cases come.
The limit value is positive infinity.
Or maybe negative infinity. We need to look at the sign and then decide.
FAQs on Understanding Limits to Infinity in Calculus
1. What does a limit to infinity mean in calculus?
A limit to infinity describes how a function behaves as x → ∞ or x → −∞, rather than approaching a specific number. It tells us the long-term behavior of a function.
- If lim (x→∞) f(x) = L, the function approaches the finite value L as x becomes very large.
- If lim (x→∞) f(x) = ∞, the function increases without bound.
- This concept is often used to determine horizontal asymptotes.
2. How do you evaluate limits as x approaches infinity?
To evaluate a limit as x approaches infinity, focus on the highest power of x in the expression. For rational functions:
- Compare the highest degree terms in the numerator and denominator.
- If degrees are equal, the limit is the ratio of leading coefficients.
- If numerator degree is smaller, the limit is 0.
- If numerator degree is larger, the limit is ∞ or −∞.
3. What is the limit of 1/x as x approaches infinity?
The limit of 1/x as x approaches infinity is 0. As x becomes very large, the fraction 1 divided by a large number becomes very small.
- lim (x→∞) 1/x = 0
- lim (x→−∞) 1/x = 0
4. What is the difference between a limit at infinity and an infinite limit?
A limit at infinity studies behavior as x becomes very large, while an infinite limit occurs when the function grows without bound near a specific value.
- Limit at infinity: lim (x→∞) f(x) = L (long-term behavior).
- Infinite limit: lim (x→a) f(x) = ∞ (vertical asymptote at x = a).
5. How do horizontal asymptotes relate to limits at infinity?
A horizontal asymptote is determined by the limit of a function as x approaches infinity. If lim (x→∞) f(x) = L, then the line y = L is a horizontal asymptote.
- Example: lim (x→∞) (5x + 1)/(x + 3) = 5.
- Therefore, the horizontal asymptote is y = 5.
6. What happens if the numerator has a higher degree than the denominator in limits to infinity?
If the numerator’s degree is higher than the denominator’s, the limit as x approaches infinity is ∞ or −∞. The function grows without bound because the numerator increases faster.
- Example: lim (x→∞) (x³)/(x²) = x → ∞.
- This means there is no horizontal asymptote.
7. How do you find the limit of a polynomial as x approaches infinity?
The limit of a polynomial as x approaches infinity is determined by its highest degree term. As x → ∞, the leading term dominates.
- Example: f(x) = 4x³ − 2x + 1
- Since the leading term is 4x³, lim (x→∞) f(x) = ∞.
- If the leading coefficient were negative, the limit would be −∞.
8. What is the limit of e^x as x approaches infinity?
The limit of e^x as x approaches infinity is ∞. The exponential function grows without bound as x increases.
- lim (x→∞) e^x = ∞
- lim (x→−∞) e^x = 0
9. Can you give an example of a limit to infinity with a worked solution?
Yes, for example lim (x→∞) (2x² + 3x)/(x² − 4) equals 2. To solve:
- Step 1: Identify highest powers (x² in numerator and denominator).
- Step 2: Divide coefficients of leading terms: 2/1.
- Step 3: Therefore, lim (x→∞) = 2.
10. What are common mistakes when solving limits to infinity?
A common mistake when solving limits to infinity is ignoring the highest degree term. The dominant term determines end behavior.
- Do not substitute ∞ directly into expressions.
- Always compare degrees for rational functions.
- Remember that lower degree terms become negligible.
- Check signs of leading coefficients for ∞ or −∞ results.
