Finding the Limit of a Function with Exponents
Whenever you are asked to find the limit of a function the first thing you should always do is plug in the value to which x is approaching.
Given lim(x→3) (x^3+3) you should plug in 3. When you do that you get 30. Therefore the limit of that function as x approaches 3 is 30.
Remember ALWAYS plug in the value that x is approaching.
If step 1 does not work then there are a couple of methods to solving limits.
If you are asked to find the limit of a function as x approaches infinity then you simply look at the exponents of the function. The limit as x approaches infinity is simply asking for the horizontal asymptote of the function.
Given lim(x→∞) (x^3+3)/(x^5+2), compare the exponents, and if the larger one is on the bottom as in this case, then the limit is equal to zero.
If the exponents are equal then just take the leading coefficients of the function and divide them.
Given lim(x→∞) (3x^3+3)/(2x^3+2), the exponents are the same so the limit is equal to 3/2 or the top leading coefficient divided by the bottoms leading coefficient.
If the larger exponent of the function is on the top then you must use what is called l'hopitals rule.
L'hopital's rule simply states that you must take the derivative of the top and of the bottom separately.
Given lim(x→∞) (3x^4+3)/(2x^3+2), take the derivative of the top which yields 12x^3 and divide it by the derivative of the bottom which is 6x^2. Then plug in infinity, and if you get infinity over infinity then you must use l'hopital's rule again. After using l'hopital's rule another 3 times you get infinity. This means that the limit does not exist.
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