Lim x → 1 નું મહત્વ શું છે?
Understand the Problem
The question is asking about the significance of the limit of x as it approaches 1, which involves understanding the behavior of a function as it gets close to a specific value.
Answer
The limit as $x$ approaches 1 is $2$.
Answer for screen readers
The limit of the function as $x$ approaches 1 is $2$.
Steps to Solve
- Identify the function
Determine the function you are analyzing as $x$ approaches 1. For example, let's say the function is $f(x) = \frac{x^2 - 1}{x - 1}$.
- Substitute the value
Substitute $x = 1$ into the function to check if it results in an indeterminate form, like $\frac{0}{0}$.
For the function $f(x) = \frac{x^2 - 1}{x - 1}$: $$ f(1) = \frac{1^2 - 1}{1 - 1} = \frac{0}{0} $$ This indicates we need to simplify further.
- Simplify the function
Factor the numerator if possible. In our case, $x^2 - 1$ factors to $(x - 1)(x + 1)$.
Thus: $$ f(x) = \frac{(x - 1)(x + 1)}{x - 1} $$ Now we can cancel the $(x - 1)$ terms (for $x \neq 1$): $$ f(x) = x + 1 $$
- Evaluate the limit
Now substitute $x = 1$ in the simplified function: $$ \lim_{x \to 1} f(x) = 1 + 1 = 2 $$
- Conclusion of the limit
Thus, the significance of the limit as $x$ approaches 1 is $2$.
The limit of the function as $x$ approaches 1 is $2$.
More Information
Finding limits is essential in calculus as it helps determine the behavior of functions at points where they may not be defined.
Tips
- Forgetting to simplify the function when encountering $\frac{0}{0}$ indeterminate forms.
- Confusing the limit with direct substitution without simplification.
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