I have an assignment question that asks: Evaluate lim (e^2x - 1)/(e^x - 1) as x tends towards zero. What is meant by evaluate?/how do i evaluate this equation?

evaluate means find the value of. The numerator is the difference of two squares. This can be factorized: A^2 - B^2 = (A-B)(A+B) So the numerator is (e^x - 1)(e^x + 1) the denominator cancels and you are left with the question: Find lim (e^x + 1) as x tends to zero e^0 is 1 so the answer is 1+1=2 In a more complex case, you have to use the definition of limit to find the answer. K is a limit of f(x) as x tends to c if for all possible positive epsilon, there exists a positive delta such that the distance of x from c being less than delta, implies that the distance of f(x) from K is less than epsilon. The proof is usually a matter of finding a formula for delta in terms of epsilon. here need to find a delta for each epsilon so that if |x| being less than delta guarantees e^x is closer than epsilon to 1. Want to guarntee: 1-epsilon < e^x < 1+epsilon take logs: ln|1-epsilon| < x < ln |1+epsilon| so let delta be the smaller of -ln|1-epsilon| and ln|1+epsilon| and x being in the slightly smaller range guarantees that x is in the needed range, and therefore so is e^x. QED!