now let's talk about how to evaluate limits using properties of limits for example let's say that the limit as x approaches a of the function f of x say that's equal to four and the limit as x approaches a of the function g of x let's say that's equal to negative 3. so using that information what is the limit as x approaches a of 4 f of x so what you're supposed to do is take the constant and move to the front so this is equivalent to four times the limit as x approaches a of f of x instinctively you know it's just four times four but if you want to show your work this is how you should write it now we can replace this expression with this value because that's what it equals to and so in the end it's 4 times 4 which is sixteen but some teachers will want you to write out every step and you just gotta do it that way here's another one what is the limit as x approaches a for this expression three f of x plus five g x let's rewrite it this is equivalent to 3 times the limit as x approaches a of f of x plus 5 times the limit as x approaches a of g of x so we know this is equal to four so what we have is three times four and this portion here is equal to negative three so that's plus five times negative three three times four is twelve five times negative three is negative fifteen and twelve minus fifteen is negative three try this one what is the limit as x approaches a of the expression f of x times g of x show your work as well now instinctively you know it's 4 times negative 3 which is negative 12. but to show your work write it like this first you want to separate f of x and g of x and then substitute so this is going to be 4 times negative 3 which is negative 12. now what about this one what is the limit as x approaches a of g of x raised to the fourth times the square root of f of x how can we rewrite the expression so first we have the limit as x approaches a of g of x which is all raised to the fourth power times the limit as x approaches a of f of x you can raise it to the one half or take the square root of the entire thing so this portion is equal to negative three so we have negative three raised to the fourth power times the square root of four negative three to the fourth power is eighty-one the square root of four is two so this is equal to one sixty-two here's another example what is the limit as x approaches a of f of x divided by g of x so how would you rewrite that expression to evaluate it here's what i would do to separate f and g i would write first the limit as x approaches a for f of x times the limit as x approaches a of one over g of x f times one over g is the same as f divided by g so we know the first part is simply equal to four the second part it's going to be one divided by negative three so the final answer is negative four over three and so now you know how to use properties of limits to evaluate limits you