Limits involving infinity: Understanding how functions behave as they approach positive or negative infinity.
Asymptotes: A line that a graph of a function approaches but never touches.
Vertical asymptote: Occurs when the limit of f(x) as x approaches a number is ±∞.
Horizontal asymptote: Occurs when the limit of f(x) as x approaches ±∞ is a real number.
Discontinuities
Removable discontinuity (Hole): Occurs when both the numerator and denominator of a rational function are zero at the same point but can be cancelled.
Non-removable discontinuity (Vertical asymptote): Occurs when the denominator is zero and cannot be cancelled out.
Identifying Discontinuities
Discontinuities occur when the denominator of a rational function is zero.
To determine if a discontinuity is a hole or an asymptote:
Hole: If you can cancel out the problematic factor (numerator and denominator both zero).
Asymptote: If you cannot cancel the problematic factor.
Example Analysis
Find where discontinuities exist in a rational function.
Example: Discontinuities at x = 1 and x = -3 are vertical asymptotes.
Determine if they are holes or asymptotes by checking if the term can be cancelled.
Horizontal Asymptotes and Limits at Infinity
If the limit of f(x) as x approaches ±∞ approaches a number, a horizontal asymptote exists.
Horizontal asymptotes are determined by the behavior of the function as x approaches infinity.
The rules for limits at infinity are the same as finite limits, but the results often involve horizontal asymptotes.
Polynomial Behavior at Infinity
Polynomials: As x approaches ±∞, polynomials tend to ±∞ based on the leading term.
Example: For x^3, as x goes to infinity, the function goes to positive infinity, and as x goes to negative infinity, the function goes to negative infinity.
The sign and behavior of the leading term dictate the end behavior of the polynomial.
Rational Functions and Limits
To evaluate limits of rational functions as x approaches infinity:
Divide the numerator and the denominator by the highest power of x in the denominator.
Evaluate the resulting expression as x approaches infinity.
Special Considerations
Absolute values in limits: When simplifying expressions involving square roots, consider absolute values to manage sign changes correctly.
Changing signs: The absolute value of x is equal to x if x is positive, and is equal to -x if x is negative.
Infinity minus infinity cases: Undefined form, requires rationalization.
Practical Examples
Root expressions: Rationalizing expressions with square roots can help in limits.
Evaluating infinity forms: Be cautious of expressions like infinity - infinity; these need special treatment.
Conclusion
Limit Techniques: Use algebraic manipulation, rationalization, and understanding of asymptotic behavior to solve limit problems as x approaches infinity.
Understanding Asymptotes: Distinguishing between horizontal and vertical asymptotes is crucial in calculus.