Determine the limit as 𝑥
approaches zero of five 𝑥 squared plus seven 𝑥 all divided by three 𝑥.
In this question, we’re asked to
evaluate the limit of a rational function. That’s the quotient of two
polynomials. And we recall we can always attempt
to evaluate the limit of a rational function by direct substitution. However, if we substitute 𝑥 is
equal to zero into our rational function and evaluate, we end up with zero divided
by zero. This is an indeterminate form. This means we can’t evaluate this
limit by direct substitution alone. We’re going to need to use some
other method. Instead, let’s try and simplify our
limit. And the first thing we could notice
is there’s a shared factor of 𝑥 in the numerator and denominator. We want to cancel this shared
factor of 𝑥 to simplify our limit.
And in fact, we’re allowed to do
this. This is because we’re taking the
limit as 𝑥 approaches zero. Remember, this means we’re
interested in what happens to the outputs of our functions as our values of 𝑥
approach zero. Therefore, the output of the
function when 𝑥 is equal to zero does not affect its limit as 𝑥 approaches
zero. Another way of thinking about this
is if 𝑥 is not equal to zero, 𝑥 divided by 𝑥 is just equal to one. Therefore, after canceling the
shared factor of 𝑥 in the numerator and denominator, we’ve rewritten our limit as
the limit as 𝑥 approaches zero of five 𝑥 plus seven all divided by three.
And now this is just a linear
function, so we can evaluate this limit by using direct substitution. We substitute 𝑥 is equal to zero
into our function to get five times zero plus seven all divided by three, which, if
we simplify, is equal to seven over three, which is our final answer. Therefore, we were able to show the
limit as 𝑥 approaches zero of five 𝑥 squared plus seven 𝑥 all divided by three 𝑥
is seven over three.