Suppose [itex]\lim_{x\to a} f(x)= \infty[/itex] and [itex]\lim_{x\to a} g(x)= \infty[/itex]. Then, given any Y> 0, there exist [itex]\delta_1> 0[/itex] such that if [itex]|x-a|< \delta_1[/itex] then f(x)> Y and there exist [itex]\delta_2< 0[/itex] such that if [itex]|x- a|< \delta_2[/itex] then g(x)> 0. Take [itex]\delta[/itex] to be the smaller of [itex]\delta_1[/itex] and [itex]\delta_2[/itex] so that if [itex]|x-a|< \delta[/itex] both [itex]|x-a|< \delta_1[/itex] and [itex]|x-a|< \delta_2[/itex] are true. Then f(x)> Y and g(x)> 0 so f(x)+ g(x)> Y+ 0.Juggler123 said:Suppose that the two functions f(x) and g(x) both tend to infinity then surely f(x) + g(x) also tends to infinity? How can you prove this though?
Similar proof. Given any real number, Y, there exist real number [itex]\delta[/itex] such that if [itex]|x-a|< \delta[/itex], [itex]f(x)> Y[/itex] and [itex]g(x)> 1[/itex]. Then f(x)g(x)> (Y)(1).Similarly f(x)*g(x) would also tend to infinity wouldn't it?
Let f(x)= g(x)= 1/(x-a). Then f and g both tend to infinity as x goes to a but f- g and f/g tend to 0 and 1 respectively.f(x) - g(x) and f(x)/g(x) wouldn't tend to anything though surely since infinity minus infinity and infinity over infinity are both undefined. Can anyone help me?
Excellent dilineation!woodyallen1 said:f=g so f-g=0 and f/g=1...
A limit of a function at infinity is the value that a function approaches as its input (usually denoted as x) becomes infinitely large. It can also be thought of as the behavior of the function as x approaches infinity.
To determine the limit of a function at infinity, we first need to identify the behavior of the function as x approaches infinity. This can be done by analyzing the highest degree term in the function. If it is a constant, the limit is simply that constant. If it is a polynomial, the limit will be either positive or negative infinity, depending on the sign of the leading coefficient. If it is a rational function, the limit may be finite or infinite, depending on the degree of the numerator and denominator.
No, a function can only have one limit at infinity. This is because the limit at infinity represents the behavior of the function as x approaches infinity, and as x can only approach infinity in one way, the limit must also be unique.
There are three types of limits at infinity: horizontal, vertical, and oblique. A horizontal limit at infinity occurs when the function approaches a constant value as x becomes infinitely large. A vertical limit at infinity occurs when the function approaches either positive or negative infinity as x becomes infinitely large. An oblique limit at infinity occurs when the function approaches a slanted line as x becomes infinitely large.
Limits at infinity can be used in many real-world applications, such as in physics, economics, and engineering. For example, in physics, limits at infinity can be used to determine the behavior of a system as time approaches infinity. In economics, they can be used to determine the long-term trends of a market. In engineering, they can be used to analyze the stability of a system as it approaches infinity.