Learn the properties of function limits complete with formulas, example problems, and how to calculate limits. Perfect for grade 11 high school students. Essential math material that often appears on exams!
🧠 What Are the Properties of Function Limits?
In grade 11 high school mathematics, the function limit is a fundamental concept that is very important for understanding calculus. To solve limit problems more quickly, we can use several properties or rules of function limits that have been mathematically proven.
📋 List of Function Limit Properties Formulas
Here are some formulas or properties of limits that are often used in solving limit problems:
1. Constant Limit
$$ \lim_{x \to a} c = c $$
Meaning, if the function is just a constant number, then its limit is also that constant.
2. Identity Function Limit
$$ \lim_{x \to a} x = a $$
3. Addition/Subtraction of Limits
$$ \lim_{x \to a} [f(x) \pm g(x)] = \lim f(x) \pm \lim g(x) $$
4. Multiplication by a Constant
$$ \lim_{x \to a} [c \cdot f(x)] = c \cdot \lim f(x) $$
5. Multiplication of Two Functions
$$ \lim_{x \to a} [f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x) $$
6. Division of Two Functions
$$ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}, \quad \text{if } \lim g(x) \neq 0 $$
7. Powers and Roots
$$ \lim_{x \to a} \left( f(x) \right)^n = \left( \lim f(x) \right)^n \quad \text{and} \quad \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim f(x)} $$
✍️ Example Problems on Function Limits and Solutions
Problem 1:
Calculate:
$$ \lim_{x \to 2} (x^2 + 2x - 1) $$
Solution:
Direct substitution:
$$ = 2^2 + 2(2) - 1 = 4 + 4 - 1 = \boxed{7} $$
Problem 2:
Calculate:
$$ \lim_{x \to 2} \frac{x^2 - 4}{x - 2} $$
Solution:
Direct substitution gives:
$$ \frac{4 - 4}{0} = \frac{0}{0} \Rightarrow \text{indeterminate form} $$
Factor the numerator:
$$ = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \Rightarrow \lim_{x \to 2} x + 2 = \boxed{4} $$
Problem 3:
Calculate:
$$ \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} $$
Solution:
Use the conjugate multiplication technique:
$$ \frac{\sqrt{x} - 3}{x - 9} \cdot \frac{\sqrt{x} + 3}{\sqrt{x} + 3} = \frac{x - 9}{(x - 9)(\sqrt{x} + 3)} = \frac{1}{\sqrt{x} + 3} $$
$$ \lim_{x \to 9} \frac{1}{\sqrt{x} + 3} = \frac{1}{3 + 3} = \boxed{\frac{1}{6}} $$
📚 Practice Problems on Function Limits
Here are some limit problems to test your understanding:
$\lim_{x \to 3} (2x^2 - x + 1)$
$\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
$\lim_{x \to 5} \frac{\sqrt{x} - \sqrt{5}}{x - 5}$
$\lim_{x \to 0} \frac{\sin x}{x}$
If $\lim_{x \to a} f(x) = 6$, and $\lim_{x \to a} g(x) = -2$, then:
$$ \lim_{x \to a} [5f(x) + 3g(x)] = ? $$
💡 Tip: Practicing these problems is very useful for school exam preparation.
📎 Conclusion
By understanding the properties of function limits, you can save a lot of time when working on math problems. You don’t always need to make tables or graphs — just use formulas and basic algebraic logic. Also, learn about indeterminate forms and how to handle them in the next section: Methods for Calculating Function Limits↝