Problems Based on Standard Results in Limits

When solving problems in calculus, particularly those involving limits, there are several standard results that are frequently used. These results are derived from the fundamental properties of limits and are essential for solving more complex problems. Understanding these standard results can simplify the process of finding limits and can be applied to a wide range of functions.

Standard Results in Limits

Here are some of the most commonly used standard results in limits:

1. $\lim_{x \to a} c = c$, where $c$ is a constant.
2. $\lim_{x \to a} x = a$.
3. $\lim_{x \to a} (x^n) = a^n$, where $n$ is a positive integer.
4. $\lim_{x \to 0} \frac{\sin x}{x} = 1$.
5. $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$.
6. $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$, where $n$ is a positive integer.
7. $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$.
8. $\lim_{x \to 0} \frac{\log(1 + x)}{x} = 1$, where $\log$ denotes the natural logarithm.

These results are often used as building blocks to solve more complicated limit problems.

Table of Differences and Important Points

Result	Condition	Importance
$\lim_{x \to a} c = c$	$c$ is a constant	Simplifies limits involving constants
$\lim_{x \to a} x = a$	-	Establishes the continuity of identity function
$\lim_{x \to a} (x^n) = a^n$	$n$ is a positive integer	Helps in polynomial limit problems
$\lim_{x \to 0} \frac{\sin x}{x} = 1$	-	Fundamental in trigonometric limits
$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$	-	Useful in trigonometric limits involving cosine
$\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$	$n$ is a positive integer	Used in derivative proofs and polynomial limits
$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$	-	Essential in exponential function limits
$\lim_{x \to 0} \frac{\log(1 + x)}{x} = 1$	-	Key in logarithmic function limits

Formulas and Examples

Example 1: Polynomial Limit

Find the limit as $x$ approaches 3 for the function $f(x) = x^2$.

Using the standard result:

$$ \lim_{x \to a} (x^n) = a^n $$

We get:

$$ \lim_{x \to 3} (x^2) = 3^2 = 9 $$

Example 2: Trigonometric Limit

Evaluate the limit as $x$ approaches 0 for the function $f(x) = \frac{\sin x}{x}$.

Using the standard result:

$$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$

We find:

$$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$

Example 3: Exponential Limit

Determine the limit as $x$ approaches 0 for the function $f(x) = \frac{e^x - 1}{x}$.

Applying the standard result:

$$ \lim_{x \to 0} \frac{e^x - 1}{x} = 1 $$

We conclude:

$$ \lim_{x \to 0} \frac{e^x - 1}{x} = 1 $$

Example 4: Logarithmic Limit

Calculate the limit as $x$ approaches 0 for the function $f(x) = \frac{\log(1 + x)}{x}$.

By using the standard result:

$$ \lim_{x \to 0} \frac{\log(1 + x)}{x} = 1 $$

We have:

$$ \lim_{x \to 0} \frac{\log(1 + x)}{x} = 1 $$

Conclusion

Understanding and applying these standard results can greatly simplify the process of evaluating limits. They are particularly useful when dealing with indeterminate forms or when the direct substitution method does not yield an immediate answer. Mastery of these results is essential for success in calculus and for solving complex problems that involve limits.