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Find .Choose answer:(A) (B) (C) (D) The limit doesn't exist
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Q. Find .Choose answer:(A) (B) (C) (D) The limit doesn't exist
- Multiply by Conjugate: To resolve the indeterminate form, we can multiply the expression by the conjugate of the denominator over itself, which is a common technique for limits involving square roots.This multiplication is valid because is equal to , and multiplying by does not change the value of the expression.
- Simplify Numerator and Denominator: Now, we simplify the numerator and the denominator separately.Numerator:Denominator:The denominator simplifies to the difference of squares:
- Distribute and Combine Like Terms: Next, we simplify the numerator by distributing to both terms inside the parentheses.Numerator:Now, we can combine like terms in the numerator:=
- Cancel Common Terms: We now have the following expression for the limit: Notice that the term in the numerator will cancel out the in the denominator when approaches , because factors to .
- Factor Out and Simplify: Let's factor out the in the denominator and cancel out the common terms:\lim_{x \to }\frac{(x)(\sqrt{x+}) + x - }{(x - )}= \lim_{x \to }(\sqrt{x+}) + Now, we can directly substitute into the simplified expression:= (\sqrt{\cdot +}) + = (\sqrt{+}) + = (\sqrt{}) + = + =
- Substitute and Calculate: Since the original expression simplifies to as approaches , the correct answer is not among the provided options , , , or The limit doesn't exist. There seems to be a mistake in the provided options.
