What is the limit of f(x)=2x^2 as x approaches 1?

Feb 7, 2015

By applying ${\lim}_{x \to 1} f \left(x\right)$, the answer to ${\lim}_{x \to 1} 2 {x}^{2}$ is simply 2.

The limit definition states that as x approaches some number, the values are getting closer to the number. In this case, you can mathematically declare that $2 {\left(\to 1\right)}^{2}$, where the arrow indicates that it approaches x = 1. Since this is similar to an exact function like $f \left(1\right)$, we can say that it must approach $\left(1 , 2\right)$.

However, if you have a function like ${\lim}_{x \to 1} \frac{1}{1 - x}$, then this statement has no solution. In hyperbola functions, depending on where x approaches, the denominator may equal zero, thus no limit at that point such exists.

To prove this, we can use ${\lim}_{x \to {1}^{+}} f \left(x\right)$ and ${\lim}_{x \to {1}^{-}} f \left(x\right)$. For $f \left(x\right) = \frac{1}{1 - x}$,

${\lim}_{x \to {1}^{+}} \frac{1}{1 - x} = \frac{1}{1 - \left(x > 1 \to 1\right)} = \frac{1}{- \to 0} = - \infty$, and
${\lim}_{x \to {1}^{-}} \frac{1}{1 - x} = \frac{1}{1 - \left(x < 1 \to 1\right)} = \frac{1}{+ \to 0} = + \infty$

These equations state that as x approaches to 1 from the right of the curve (${1}^{+}$), it keeps going down infinitely, and as x approaches from the left of the curve (${1}^{-}$), it keeps going up infinitely. Since these two parts of x = 1 do not equal, we conclude that ${\lim}_{x \to 1} \frac{1}{1 - x}$ does not exist.

Here is a graphical representation:
graph{1/(1-x) [-10, 10, -5, 5]}
Overall, when it comes to limits, make sure to watch for any equation that has a zero in the denominator (including others like ${\lim}_{x \to 0} \ln \left(x\right)$, which does not exist). Otherwise you will have to specify if it approaches zero, infinity, or -infinity using the notations above. If a function is similar to $2 {x}^{2}$, then you can solve for it by substituting x into the function using the limit definition.

Whew! It sure is a lot, but all the details are very important to note for other functions. Hope this helps!