Limit Name Log For Free

here's a question for you how would you evaluate this limit what is the limit as n goes to infinity of a 1 plus 1 over n raised to the n power well in this video we're going to talk about how to evaluate this limit so the first thing we're going to do is we're going to set the entire expression equal to y so our goal is to solve for y if we can do that we could find the value of this expression now the next thing that we're going to do is we're going to take the natural log of both sides so we're going to have ln y is equal to the limit as n goes to infinity and then the natural log of 1 plus 1 over n raised to the n now a property of logs let's say we have log x raised to the a we can take the exponent and move it to the front so ln x raised to the a power is a times ln x so we're going to take this variable move it to the front so we're going to have the natural log of y is equal to the limit as n goes to infinity n times the natural log of one plus one over n now we still have some more work to do here right now we have the product of two expressions n times ln one plus one over n we need to convert this expression from being a product of two expressions to a quotient of two expressions and here's how we can do that for instance a times b is equal to b divided by one over a so here we have a product of two expressions now we have a quotient of two expressions you can illustrate this with real numbers four times five which is twenty is equivalent to five divided by one fourth they both equal twenty so based on that we're going to convert this expression from n times the natural log of 1 plus 1 over n to ln 1 plus 1 over n divided by 1 over n the mathematically equivalent so now in this format we could use something called a l'hopital's rule which is useful for evaluating limits that approach infinity so we're going to take the derivative of the top and of the bottom let's start with the numerator the derivative of the natural log of u is equal to u prime over u so since the u variable is going to be the stuff inside of the natural log function that's going to be 1 plus 1 over n u prime is going to be the derivative of that expression the derivative of 1 is 0. now the derivative of 1 over n first we need to rewrite that as n to the minus one and then we need to use the power rule so it's going to be negative one n to the minus two and then we can rewrite that as negative one over n squared so using the formula u prime over u we're going to have ln y is equal to the limit as n goes to infinity and the derivative of that expression is going to be u prime which is negative 1 over n squared divided by u which is what we see here that's 1 plus 1 over n and all of that is divided by the derivative of one over n which is this again negative one over n squared whoa what just happened now let's get rid of that now the next thing that we need to do is cancel the two negative signs a negative divided by a negative sign is a positive sign and then we can...