A limit as x approaches infinity is the value that a function approaches as the input variable, x, gets closer and closer to infinity. It is denoted by the symbol "lim" followed by the function, with an arrow pointing towards the value of infinity.
To solve a limit problem as x approaches infinity, you can follow these steps:
1. Simplify the expression by factoring and canceling out common terms.
2. Plug in infinity for the variable x.
3. If the resulting expression is an indeterminate form (such as 0/0 or ∞/∞), use algebraic manipulation or L'Hôpital's rule to simplify it.
4. If the resulting expression is a finite number, that is the limit as x approaches infinity. If the expression is infinity, the limit does not exist.
Some common types of limit problems as x approaches infinity include:
1. Rational functions (polynomial functions with a fraction in the form of ax^m / bx^n)
2. Exponential functions (in the form of a^x where a is a constant)
3. Logarithmic functions (in the form of log(x) or ln(x))
4. Trigonometric functions (in the form of sin(x) or cos(x))
Yes, a limit as x approaches infinity can be negative. The sign of the limit depends on the behavior of the function as x gets closer and closer to infinity. If the function approaches a negative value, the limit will be negative. If the function approaches a positive value, the limit will be positive.
A one-sided limit as x approaches infinity only considers the behavior of the function as x approaches infinity from one direction (either positive or negative). A two-sided limit as x approaches infinity considers the behavior of the function from both directions and requires that the function approaches the same value from both sides in order for the limit to exist.