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Q.
- Identify the form: Identify the form of the limit.We need to determine the form of the limit to decide the appropriate method to solve it. Let's substitute into the function to see if it results in an indeterminate form.Substitute into the function:= = = = = This is an indeterminate form, so we need to use algebraic manipulation to simplify the expression and find the limit.
- Simplify using algebraic manipulation: Simplify the expression using algebraic manipulation.To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of is .Multiply the original expression by the conjugate over itself:This will help us eliminate the square root in the numerator.
- Perform the multiplication: Perform the multiplication.Now, we multiply the numerators and the denominators separately.Numerator:Denominator:We don't need to multiply this out because we are looking for the limit as approaches , and we are interested in whether the denominator will cancel out the zero in the numerator.
- Simplify further: Simplify the expression further.Now that we have the simplified numerator, we can see if it cancels with the denominator.Numerator: Denominator: Notice that the numerator can be factored as .Numerator after factoring: Now, we can cancel out the term in the numerator and the denominator.
- Cancel common terms: Cancel out the common terms and find the limit.After canceling out the term, we are left with:Numerator: Denominator: Now, we can substitute into the remaining expression to find the limit.Limit as approaches :So, the limit of the function as approaches is .
