Math Problem Statement

Evaluate the following: (i) lim x→1− 2^1/(x−1), (ii) lim x→0 (√(4+x)−4)/x, (iii) lim x→2 (x^2−4)/(x+2), (iv) lim x→a (sin(x)−sin(a))/(x−a)

Solution

Let's evaluate each limit step by step.

(i) limx121x1\lim_{x \to 1^-} 2^{\frac{1}{x-1}}

As x1x \to 1^-, the exponent 1x1\frac{1}{x-1} tends towards -\infty, since x1x - 1 approaches a small negative value, meaning the denominator approaches 0 from the negative side.

Thus, the expression becomes 22^{-\infty}, which tends to 0.

0\boxed{0}

(ii) limx04+x4x\lim_{x \to 0} \frac{\sqrt{4+x} - 4}{x}

To solve this, we rationalize the numerator by multiplying and dividing by the conjugate:

4+x4x4+x+44+x+4=(4+x)16x(4+x+4)=x12x(4+x+4)\frac{\sqrt{4+x} - 4}{x} \cdot \frac{\sqrt{4+x} + 4}{\sqrt{4+x} + 4} = \frac{(4+x) - 16}{x(\sqrt{4+x} + 4)} = \frac{x - 12}{x(\sqrt{4+x} + 4)}

Simplifying further:

=124+x+4= \frac{-12}{\sqrt{4+x} + 4}

Now, as x0x \to 0:

124+0+4=124+4=128=32\frac{-12}{\sqrt{4+0} + 4} = \frac{-12}{4+4} = \frac{-12}{8} = -\frac{3}{2}

Thus, the limit is:

32\boxed{-\frac{3}{2}}

(iii) limx2x24x+2\lim_{x \to 2} \frac{x^2 - 4}{x + 2}

First, factor the numerator:

x24x+2=(x2)(x+2)x+2\frac{x^2 - 4}{x + 2} = \frac{(x - 2)(x + 2)}{x + 2}

For x2x \neq -2, cancel the x+2x + 2 terms:

=x2= x - 2

Now, as x2x \to 2:

22=02 - 2 = 0

Thus, the limit is:

0\boxed{0}

(iv) limxasinxsinaxa\lim_{x \to a} \frac{\sin x - \sin a}{x - a}

This limit is in the standard form of the derivative of the sine function. The derivative of sinx\sin x at x=ax = a is:

cosa\cos a

Thus, the limit is:

cosa\boxed{\cos a}


Do you want further details or have any questions?

Here are five related questions you might explore:

  1. How do you evaluate limits involving logarithms?
  2. What is the limit of exponential functions as the base approaches 0?
  3. How does rationalizing the numerator help in solving limits?
  4. What is L'Hopital's rule and when can it be applied?
  5. How do you find the derivative of trigonometric functions?

Tip: Always check if a limit problem involves an indeterminate form (like 00\frac{0}{0}), which may suggest factoring, conjugation, or applying L'Hopital's rule for simplification.

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Math Problem Analysis

Mathematical Concepts

Limits
Exponential Functions
Rationalizing
Derivatives
Trigonometric Limits

Formulas

lim x→a (f(x) - f(a))/(x - a) = f'(a) (definition of derivative)
Standard limit lim x→a (sin(x) - sin(a))/(x - a) = cos(a)

Theorems

Limit laws
Derivative of sine function

Suitable Grade Level

Undergraduate Calculus