Question

Please help me with these

Answer 1

When we approach limits, we are finding values that are infinitesimally approaching this x-value. Essentially, we consider the approximate location that this root or limit appears. This is essential when it comes to taking Calculus, and finding the limit or rate of change of a function.

When we are attempting limits questions, there are several tests we attempt first.

1. Evaluate the limit by substituting the value of the x-value as it approaches the value (direct evaluation of a limit)
2. Rearrangement of the function, such that we can evaluate the limit.
3. (TRIGONOMETRIC PROPERTIES)
$\lim_{x \to 0} (\frac{sinx}{x}) = 1$
$\lim_{x \to 0} (\frac{tanx}{x}) = 1$
4. Using L'Hopital's Rule for indeterminate limits, such as 0/0, -infinity/infinity, or infinity/infinity.

For example:

1) $\lim_{x \to 0}\frac{\sqrt{x} - 5}{x - 25}$

We can do this using the first and second method.
Method 1: Direct evaluation:

Substitute x = 0 to the function.
$\frac{\sqrt{0} - 5}{0 - 25}$
$= \frac{-5}{-25}$
$= \frac{1}{5}$

Method 2: Rearranging the function

We can see that x - 25 can be rewritten as: (√x - 5)(√x + 5)
By rewriting it in this form, the top will cancel with the bottom easily, and our limit comes out the same.

$\lim_{x \to 0}\frac{(\sqrt{x} - 5)}{(\sqrt{x} - 5)(\sqrt{x} + 5)}$
$= \lim_{x \to 0}\frac{1}{(\sqrt{x} + 5)}}$
$= \frac{1}{5}$

Every example works exactly the same way, and by remembering these criteria, every limit question should come out pretty naturally.