Limit as x approaches a

Definition

The limit as x approaches a refers to the value that a function f(x) gets closer to as the input x approaches the value a. This concept is essential in calculus as it helps in understanding the behavior of functions near specific points, especially when those points may not be directly evaluated due to discontinuities or undefined values. Recognizing limits also aids in defining derivatives and integrals, fundamental concepts in calculus.

5 Must Know Facts For Your Next Test

1. Limits can be approached from both sides, and if both sides yield the same value, the limit exists.
2. If a function has a removable discontinuity at x = a, the limit can still exist even though f(a) is not defined.
3. Limits can also approach infinity, indicating that as x gets closer to a certain value, f(x) increases or decreases without bound.
4. The notation $$ ext{lim}_{x o a} f(x)$$ represents the limit of f(x) as x approaches a.
5. Understanding limits is crucial for calculating derivatives using the definition of the derivative as a limit.

Review Questions

• Explain how to determine if a limit exists as x approaches a from both sides.
  • To determine if a limit exists as x approaches a from both sides, you need to evaluate the function f(x) as x gets closer to a from the left (denoted as $$ ext{lim}_{x o a^-} f(x)$$) and from the right (denoted as $$ ext{lim}_{x o a^+} f(x)$$). If both one-sided limits yield the same value, then we say that the limit exists and can be written as $$ ext{lim}_{x o a} f(x)$$. If they yield different values or at least one does not exist, then the overall limit does not exist.
• Discuss how removable discontinuities affect limits and give an example.
  • Removable discontinuities occur when there is a hole in the graph of a function at x = a, usually because f(a) is undefined or does not equal the limit. For example, consider the function f(x) = $$rac{x^2 - 1}{x - 1}$$ which has a removable discontinuity at x = 1 since it simplifies to f(x) = x + 1 for all x ≠ 1. The limit exists and equals 2 as x approaches 1, even though f(1) is undefined. Thus, while the function does not have a value at this point, we can still find that its limit exists.
• Evaluate how limits are foundational for defining derivatives and provide an example.
  • Limits are foundational for defining derivatives because the derivative of a function at a point is defined as the limit of the average rate of change of that function over an interval as the interval shrinks to zero. Mathematically, this is expressed as $$f'(a) = ext{lim}_{h o 0} rac{f(a+h) - f(a)}{h}$$. For instance, for f(x) = x^2, we can calculate its derivative at x = 2 by finding $$f'(2) = ext{lim}_{h o 0} rac{(2+h)^2 - 4}{h}$$. By simplifying and taking the limit, we find that f'(2) = 4, demonstrating how limits are used to derive rates of change.

Related terms

Continuous Function: A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.

One-Sided Limit: A one-sided limit refers to the limit of a function as the variable approaches a certain value from one side, either the left or the right.

Infinite Limit: An infinite limit occurs when the values of a function grow without bound as the input approaches a certain value.