If you're looking for help with calculus limits, you've come to the right place! Here, we provide comprehensive guidance and examples to help you understand this fundamental concept.
Basic Concepts
- Limit: The value that a function approaches as the input approaches a certain value.
- Direct Substitution: A method to find the limit of a function by substituting the value of the input into the function.
- Indeterminate Forms: Situations where direct substitution results in an undefined expression like 0/0 or ∞/∞.
Common Limit Rules
- Constant Rule: The limit of a constant function is the constant itself.
$$ \lim_{x \to a} c = c $$ - Identity Rule: The limit of a function is the function itself.
$$ \lim_{x \to a} x = a $$ - Power Rule: The limit of x^n as x approaches a is a^n.
$$ \lim_{x \to a} x^n = a^n $$ - Sum and Difference Rule: The limit of a sum or difference of functions is the sum or difference of their limits.
$$ \lim_{x \to a} (f(x) ± g(x)) = \lim_{x \to a} f(x) ± \lim_{x \to a} g(x) $$ - Product Rule: The limit of a product of functions is the product of their limits.
$$ \lim_{x \to a} f(x)g(x) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) $$ - Quotient Rule: The limit of a quotient of functions is the quotient of their limits.
$$ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} $$
Example
Find the limit of the following expression: $$ \lim_{x \to 2} \frac{x^2 - 4}{x - 2} $$
To solve this, we can factor the numerator and cancel the common factor in the denominator: $$ \lim_{x \to 2} \frac{(x + 2)(x - 2)}{x - 2} $$ $$ \lim_{x \to 2} (x + 2) $$ $$ \lim_{x \to 2} 4 = 4 $$
For more in-depth examples and explanations, check out our Calculus Limits Tutorial.
Calculus