Determine what type of function you are studying. A rational function has the form y(x) = f(x)/g(x), where f and g are distinct polynomial equations. Plug in increasing values for x. If the degree of f(x) is higher than the degree of g(x), you will see that the right-hand limit (limit as x approaches positive infinity) and left-hand limit (limit as x approaches negative infinity) will be either positive or negative infinity. If, however, the degree of g(x) is higher than the degree of f(x), both the left-hand and right-hand infinite limit will equal 0. Since the denominator has a higher degree, its absolute value will increase faster than the numerator's, pulling the value down to zero as x approaches infinity.
Calculate the ratio of the leading coefficients if f(x) and g(x) have the same degree. For the rational equation f(x) = (3x^3+10x+5) / (-2x^3-2), this will be -3/2. The right-hand and left hand are both -3/2, because as x gets arbitrarily large in either direction the leading term takes precedence over the other terms.
Write the right-hand and left-hand limits in the following manner once you have found them:
lim(x'∞) f(x) = right-hand; lim(x'-∞) f(x) = left-hand
(replace "right-hand" and "left-hand" with their respective values)
Determine whether the infinite limit of your trigonometric equation oscillates. The sine and cosine function are both oscillating because they go between -1 and 1 indefinitely. The only trig function with defined infinite limits is the arctan function: A*arctan(Bx) + C. To find the limit, calculate 2Aπ / 2 for the right-hand limit and -2Aπ / 2 for the left-hand limit.
If the function is not rational or trigonometric, it may be exponential. Determine whether the equation is of the form f(x) = Ae^(Bx) + C (where A, B and C are real numbers). If so, the limit will depend on whether A is positive or negative. If A is positive, there is no right-hand limit, and the left hand limit is equal to C. If A is negative, there is no left hand limit, and the right-hand limit is equal to C.
Pull out your graphing calculator. For other equations, such as logistical equations, you can determine the infinite limit by graphing f(x) and calculating y-values when x = 10, 50, 100, 1000, and then -10, -50, -100, -1000. Eventually the y-values will begin to converge toward a number, which corresponds to the limit. If the y-values don't converge to a number, but continue to increase or decrease, the limit is positive or negative infinity.