Evaluate the following limit: $\lim_{x \to 0} \frac{(1+x)^4 - 1}{x}$
Understand the Problem
The question asks to evaluate the limit of the expression ((1+x)^4 - 1) divided by (x) as (x) approaches 0. This involves concepts of calculus, specifically limits and possibly L'Hôpital's rule or binomial expansion.
Answer
4
Answer for screen readers
4
Steps to Solve
- Expand the binomial term
Expand $(1+x)^4$ using the binomial theorem: $$(1+x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4$$
- Substitute the expansion into the limit expression
Substitute the expanded form into the limit expression: $$ \lim_{x \to 0} \frac{(1+4x+6x^2+4x^3+x^4) - 1}{x} $$
- Simplify the expression
Simplify the numerator by canceling the 1's: $$ \lim_{x \to 0} \frac{4x+6x^2+4x^3+x^4}{x} $$
- Factor out x from the numerator
Factor out an $x$ from the numerator: $$ \lim_{x \to 0} \frac{x(4+6x+4x^2+x^3)}{x} $$
- Cancel out the x terms
Cancel out the common factor $x$: $$ \lim_{x \to 0} (4+6x+4x^2+x^3) $$
- Evaluate the limit
Evaluate the limit by substituting $x = 0$: $$ 4 + 6(0) + 4(0)^2 + (0)^3 = 4 $$
4
More Information
The limit of the given expression as $x$ approaches 0 is 4. This result can also be interpreted as the derivative of $f(x) = (1+x)^4$ evaluated at $x=0$.
Tips
A common mistake would be attempting to directly substitute $x=0$ into the original expression, which leads to an indeterminate form of $\frac{0}{0}$. Therefore, algebraic manipulation is necessary before evaluating the limit. Additionally, errors in binomial expansion could lead to an incorrect limit.
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