Consider the limit of the rational function pxlqx What concl

Consider the limit of the rational function p(x)lq(x). What conclusion can you make if direct substitution produces each expression? Write a short paragraph explaining your results. lim_x rightarrow c p(x)/q(x) = 0/1 lim_x rightarrow c p(x)/q(x) = 1/1 lim_x rightarrow c p(x)/q(x) = 1/0 lim_x rightarrow c p(x)/q(x) = 0/0

Solution

a) If direct substitution results in 0/1 form, then limit will exist and it will simply be 0.

b) If direct substitution results in 1/1 form, then limit will exist and it will simply by 1.

c) If direct substitution results in 1/0 form, then we need to check the left and right hand limists. Mostly, limit doesn\'t exist when we get 1/0 from on direct substitution.

d) If direct substitution results in 0/0 from, then we cannot predict the limit directly, we need to either simplify the given rational expression before retrying the substitution or we need to apply L\'hopitals rule. Using L\'hopitals rule we simply derivative the numerator and denominator separately before retrying the substitution. If on substitution, we get 0/0 from again, apply L\'hopitals rule again. We might have to apply L\'hopitals rule couple of times before we get a the value of limit. This kind of form (0/0) is known as indeterminate form.