Question

What is the limit of \[f\left( x \right)\] as x approaches 0?

Answer 1

Hint: In this problem, we have to find the limit of \[f\left( x \right)\], when x approaches 0. We can see that the limit of \[f\left( x \right)\] as x approaches 0 will be undefined in terms when the denominator is 0., since both sides approach different values. It depends on the function we use. We can first take an example function and we can evaluate the limit for \[x \to 0\] and check for the answer to find what happens when x approaches 0.

Complete step by step solution:
Here we have to find the limit of \[f\left( x \right)\] as x approaches 0.
We should know that the limit of the function depends upon the type of the function.
We can now take an example function to check what happens when x approaches 0.
We can now take \[f\left( x \right)=\dfrac{1}{x}\]
We can now apply the limit for x, as it approaches 0, we get
\[\Rightarrow \displaystyle \lim_{x \to {{0}^{+}}}=\dfrac{1}{x}=\dfrac{1}{0}=+\infty \]
We can see that the function where x approaches 0, becomes enormous in a positive direction.
We can now take numbers from the left, we get
\[\Rightarrow \displaystyle \lim_{x \to {{0}^{-}}}=\dfrac{1}{x}=\dfrac{1}{0}=-\infty \]
We can see that the function where x approaches 0, becomes enormous in a negative direction.
We can now take another example function \[f\left( x \right)=4x+1\]
We can now apply the limit for x, as it approaches 0, we get
\[\Rightarrow \displaystyle \lim_{x \to 0}=0+1=1\]
Where \[f\left( x \right)\] tends to 1, when x approaches infinity.
Therefore, the limit of the function depends on the function, when x approaches 0.

Note: We should always remember that the limit of the function depends on the function, when x approaches 0 or any other limits we apply. We should also know how to apply limits to the function by evaluating the limit value in the given function and simplifying it to find the value of \[f\left( x \right)\].